7^th June 2010 / Session 1.3

Last Look at Long Run Interpretation / Perfect Samples

From here: http://www.mindspring.com/~cjalverson/_1sthourlyfall2008_VBKey.htm

Case Two | Long Run Argument, Perfect Samples | Birthweight

Low birthweight is a strong marker of complications in liveborn infants. Low birthweight is strongly associated with a number of complications, including infant mortality, incomplete and impaired organ development and a number of birth defects. Suppose that the following probability model applies to year 2005 United States Resident Live Births:

Birthweight Status	Probability
Very Low Birthweight (<1500g)	.016
Low Birthweight (1500g ≤ Birthweight < 2500g)	.067
Full Birthweight (≥ 2500g)	.917
Total	1.00

Each row of the model yields a statement about an event within a population.

Interpret each probability using the Long Run Argument.

Clearly specify both the event and the population in an indefinite random sampling context.

In long runs of random sampling of US resident Live Births during year 2005, approximately 1.6% of sampled births present birthweights strictly below 1500 grams.

In long runs of random sampling of US resident Live Births during year 2005, approximately 6.7% of sampled births present birthweights of 1500 grams or greater, but strictly below 2500 grams.

In long runs of random sampling of US resident Live Births during year 2005, approximately 91.7% of sampled births present birthweights of 2500 grams or greater.

Compute and discuss Perfect Samples for n=2000.

Show full detail in computing an expected count for each event in the model.

Very Low Birthweight: E_VLB = 2000*Pr{VLB} = 2000*0.016 = 32

Low Birthweight: E_LB = 2000*Pr{LB} = 2000*0.067 = 134

Full Birthweight: E_FB = 2000*Pr{FB} = 2000*0.917 = 1834

Clearly specify both the event and the population in the specific random sampling context.

In random samples of 2000 US resident Live Births during year 2005, approximately 32 of the sampled births present birthweights strictly below 1500 grams.

In random samples of 2000 US resident Live Births during year 2005, approximately 134 of sampled births present birthweights of 1500 grams or greater, but strictly below 2500 grams.

In random samples of 2000 US resident Live Births during year 2005, approximately 1834 of sampled births present birthweights of 2500 grams or greater.

Probability Rules

Computing Probabilities Algebraically

A Probability function Pr{*} is a relationship between a process or population and the real interval [0,1].

Domain: D{Pr}= “The set of events associated with a process or a population.”

Range: R{Pr} = “The set of probabilities associated with the events in D{Pr}.”

For each event E in D{Pr}, there is a value P_E=Pr{E}, called the probability for event E.

Pr: D(Event Space) ® PS(Probability Space)

Any event E with Pr{E} = 0 is impossible.

Events E with Pr{E} @ 0 are rare.

Any event E with Pr{E} = 1 is certain.

Probability Rules: A Fair, Six-sided Die

Begin with a die with six sides: 1,2,3,4,5,6. Suppose that this die is fair - that each face has an equal chance of showing in tosses of the die. From earlier discussions, this table shouldn't require much explanation:

Face Value	Probability (Proportion)	Probability (Percentage)
1	1/6	100*(1/6) » 16.67%
2	1/6	100*(1/6) » 16.67%
3	1/6	100*(1/6) » 16.67%
4	1/6	100*(1/6) » 16.67%
5	1/6	100*(1/6) » 16.67%
6	1/6	100*(1/6) » 16.67%
Total	6/6	100*(6/6) » 100%**

The Fair d6 Model

FV: Face Values: 1, 2, 3, 4, 5, 6

Fair Model: Equally likely face values – 1/6 per face value

Pr{d6 Shows 1} = (1/6) @ .1667 or 16.67%

In long runs of tosses, approximately 1 toss in 6 shows “1”.

Pr{d6 Shows 2} = (1/6) @ .1667 or 16.67%

In long runs of tosses, approximately 1 toss in 6 shows “2”.

Pr{d6 Shows 3} = (1/6) @ .1667 or 16.67%

In long runs of tosses, approximately 1 toss in 6 shows “3”.

Pr{d6 Shows 4} = (1/6) @ .1667 or 16.67%

In long runs of tosses, approximately 1 toss in 6 shows “4”.

Pr{d6 Shows 5} = (1/6) @ .1667 or 16.67%

In long runs of tosses, approximately 1 toss in 6 shows “5”.

Pr{d6 Shows 6} = (1/6) @ .1667 or 16.67%

In long runs of tosses, approximately 1 toss in 6 shows “6”.

Basic Events

In repeated tosses of our die, the most basic possible outcomes are the faces themselves - the individual face values are the basic events. Each basic event has the same probability - (1/6).

Additive Rule: first, write down the simple events which form the event. Then, add the probabilities for each of those simple events - this total is the probability for the event.

Pr{Event} = Pr{One of the Items in the List Occurs} = Pr{1^st List Item} + … + Pr{Last List Item}

Define the event EVEN as follows: "an even face (2,4,6) shows". Then the probability of the event EVEN can be computed as

Pr{EVEN} = Pr{ exactly one of 2 or 4 or 6 shows } = Pr{2 shows} + Pr{4 shows} + Pr{6 shows} = (1/6) + (1/6) + (1/6) = 3/6 = .50

Complementary Rule: first, write down the opposite of the event. Next, write down the simple events which form the opposite event. Then, add the probabilities for each of those simple events - this total is the probability for the opposite event. Finally, subtract the probability for the opposite event from 1. The result of this subtraction is the probability for the original event.

Event=E

Opposite Event = ~E

Compute Pr{~E}

Then compute Pr{E} = 1 - Pr{~E}

Define the event 2PLUS as "a face greater than or equal to 2 shows". Then its complementary event is Not2PLUS is "a face strictly less than 2 shows", and can be computed as

Pr{not2PLUS} = Pr{ 1 shows } = 1/6.

Then compute the probability for the event 2PLUS as :

Pr{2PLUS} =  1 - Pr{not2PLUS} = 1 - (1/6) = 5/6 ≈ .8333 or as  83.33%.

Working directly,

Pr{2PLUS} =  

Pr{one of 2,3,4,5,6 shows} =

Pr{2 shows}+ Pr{3 shows}+ Pr{4 shows}+ Pr{5 shows}+ Pr{6 shows} =

(1/6) + (1/6) + (1/6) + (1/6) + (1/6) = 5/6 ≈ .8333 or as 83.33%.

A Color Sequence Experiment

Suppose that we have a special box - each time we press a button on the box, it prints out a sequence of colors, in order - it prints four colors at a time. Suppose the box follows the following Probabilities for each Color Sequence:

The Model

Color Sequence (CS)	Probability (Proportion | Percent)
BBBB	.10 | 100*(.10) = 10%
BGGB	.25 | 100*(.25) = 25%
RGGR	.05 | 100*(.05) = 05%
YYYY	.30 | 100*(.30) = 30%
BYRG	.15 | 100*(.15) = 15%
RYYB	.15 | 100*(.15) = 15%
Total	1.00 | 100*(1.00) = 100%

Let's define the experiment: We push the button, and then the box prints out exactly one (1) of the above listed color sequences. We then note the resulting (printed out) color sequence.

Let's discuss the simple (or basic) events.

The simple events are the color sequences. The probabilities for each color sequence are given in the table.

Suppose we define the event E={Blue(B) is printed in the 2^nd or 3^rd slot}. Compute the probability for event E, and show me how you did it. Also, interpret the probability for event E.

The only color sequence meeting the definition of E is the sequence BBBB.

So, we write Pr{E} = Pr{BBBB} = .10 = 10%.

This means that in long runs of box-prints, that approximately 10% of the prints will show as BBBB.

Suppose we define the event F={Yellow is printed at least once in the sequence}. Compute the probability for event F, and show me how you did it. Also, interpret the probability for event F.

The event F merely requires that Yellow be present.

Yellow is present in the following color sequences: YYYY, BYRG and RYYB.

So we write

Pr{F} =

Pr{exactly one of YYYY, BYRG or RYYB is printed} =

Pr{ YYYY }+Pr{ BYRG }+Pr{ RYYB } =

.30+.15+.15 = .60 = 60%

In long runs of box-prints, approximately 60% of prints will contain at least one Yellow in the color sequence.

Suppose we define the event G={Green(G) is printed in the 2^nd slot}. Compute the probability for event G, and show me how you did it. Also, interpret the probability for event G.

The event G requires that Green be present in the 2^nd slot.

This requirement is met in the following color sequences: BGGB, RGGR.

So we write

Pr{G} =

Pr{exactly one of BGGB or RGGR is printed} =

Pr{ BGGB }+Pr{ RGGR =

.25+.05 = .30 = 30%

In long runs of box-prints, approximately 30% of prints will show green in the 2^nd slot.

Suppose we define the event H={Red is not the 1^st color}. Compute t|he probability for the event H, and use the Complementary Rule. Also, interpret the probability for H.

The event notH requires that the sequence lead off with Red.

This requirement is met in the following color sequences: RGGR, RYYB.

Pr{notH} =

Pr{Red is the first color } =

Pr{exactly one of RGGR or RYYB is printed } =

Pr{ RGGR }+Pr{ RYYB } = .05 + .15 = .20 = 20%.

So, Pr{H} = 1 - Pr{notH} = 1 - .20 = .80 = 80%

So in long runs of box-prints, approximately 80% of color sequences will not show Red as the first color in the sequence.

Suppose we define the event I={Blue(B) is not the 4^th color}. Compute the probability for the event I, and use the Complementary Rule. Also, interpret the probability for I.

The event notI requires that the sequence end with Blue.

This requirement is met in the following color sequences: BBBB, BGGB and RYYB.

Pr{notI} =

Pr{Blue is the 4^th color } =

Pr{exactly one of BBBB, BGGB or RYYB is printed } =

Pr{BBBB}+Pr{ BGGB }+Pr{ RYYB } =

.10 + .25 + .15 = 50%.

So, Pr{I} = 1 - Pr{notI} = 1 - .50 = .50 = 50%

So in long runs of box-prints, approximately 50% of color sequences will not show Blue as the last color in the sequence.

Events and Random Variables

Color Sequence (CS)	Probability (Proportion | Percent)
BBBB	.10 | 100*(.10) = 10%
BGGB	.25 | 100*(.25) = 25%
RGGR	.05 | 100*(.05) = 05%
YYYY	.30 | 100*(.30) = 30%
BYRG	.15 | 100*(.15) = 15%
RYYB	.15 | 100*(.15) = 15%
Total	1.00 | 100*(1.00) = 100%

Consider the random variable Blue Count, defined as the number of blue slots showing in the sequence. Blue Count groups the color sequences based on common value.

Blue Count	Probability (Proportion | Percent)
4 {BBBB}	.10 | 100*(.10) = 10%
2 {BGGB}	.25 | 100*(.25) = 25%
0 {RGGR, YYYY}	.05 + .30 = .35 | 35%
1 {BYRG, RYYB}	.15 + .15 = .30 | 30%
Total	1.00 | 100*(1.00) = 100%

Blue Count induces four events: Blue Count = 0, Blue Count = 1, Blue Count = 2, Blue Count = 4. If you’re being picky, you might include an empty event for Blue Count = 3.

Pr{Blue Count = 0} = Pr{One of RGGR or YYYY Shows} = Pr{RGGR} + Pr{YYYY} = .05 + .30 = .35

Pr{Blue Count = 1} = Pr{One of BYRG or RYYB Shows} = Pr{BYRG} + Pr{RYYB} = .15 + .15 = .30

Pr{Blue Count = 2} = Pr{ BGGB } = .25

Pr{Blue Count = 3} = Pr{No qualifying sequences} = 0

Pr{Blue Count = 4} = Pr{BBBB} = .10

Color Sequence (CS)	Probability (Proportion | Percent)
BBBB	.10 | 100*(.10) = 10%
BGGB	.25 | 100*(.25) = 25%
RGGR	.05 | 100*(.05) = 05%
YYYY	.30 | 100*(.30) = 30%
BYRG	.15 | 100*(.15) = 15%
RYYB	.15 | 100*(.15) = 15%
Total	1.00 | 100*(1.00) = 100%

Consider the random variable Green Count, defined as the number of green slots showing in the sequence. Green Count groups the color sequences based on common value.

Color Sequence (CS)	Probability (Proportion | Percent)
0 {BBBB,YYYY, RYYB}	.10 + .30 + .15 = .55| 55%
2 {BGGB, RGGR}	.25 + .05 = .30 | 30%
1 {BYRG}	.15 | 100*(.15) = 15%
Total	1.00 | 100*(1.00) = 100%

Grren Count induces three events: Green Count = 0, Green Count = 1and Green Count = 2.

Pr{Green Count = 0} = Pr{One of BBBB, YYYYor RYYB Shows} = Pr{ BBBB} + Pr{YYYY} +

Pr{RYYB} = .10 + .30 + .15 = .55

Pr{Green Count = 1} = Pr{BYRG} = .15

Pr{Blue Count = 2} = Pr{One of BGGB or RGGR Shows} = Pr{BGGB} + Pr{RGGR} = .25 + .05 = .30

Pairs of Dice and Random Variables.

Case Study #1.7: Pairs to Sums

Case Description: Work with a random variable that acts on pairs of outcomes.

We assume that the dice are fair, and that the dice operate separately and independently.

Case Study Objectives:

We toss a pair of fair dice, one three-sided d3:(faces 1,2,3) and one four-sided d4:(faces 1,2,3,4).

How many pairs are possible, and what is the probability for each pair ?

Fair D4 model 

Face Value	Probability
1	1/4
2	1/4
3	1/4
4	1/4
Total	4/4

Fair D3 model

Face Value	Probability
1	1/3
2	1/3
3	1/3
Total	3/3

There are 4*3=12 distinct pairs possible: Writing each pair as (d4 face value, d3 face value):

(1,1), (2,1), (3,1), (4,1), (1,2), (2,2), (3,2), (4,2), (1,3), (2,3), (3,3), (4,3)

(d4,d3)	1	2	3	4
1	(1,1)	(2,1)	(3,1)	(4,1)
2	(1,2)	(2,2)	(3,2)	(4,2)
3	(1,3)	(2,3)	(3,3)	(4,3)

Under the independent multiplication principle,

Pr{(event from d4, event from d3)} = Pr{ event from d4}*Pr{ event from d3}, so:

Pr{(1,1)} = Pr{1 shows from d4}*Pr{1 shows from d3} = (1/4)*(1/3) = 1/12

Pr{(2,1)} = Pr{2 shows from d4}*Pr{1 shows from d3} = (1/4)*(1/3) = 1/12

Pr{(3,1)} = Pr{3 shows from d4}*Pr{1 shows from d3} = (1/4)*(1/3) = 1/12

Pr{(4,1)} = Pr{4 shows from d4}*Pr{1 shows from d3} = (1/4)*(1/3) = 1/12

Pr{(1,2)} = Pr{1 shows from d4}*Pr{2 shows from d3} = (1/4)*(1/3) = 1/12

Pr{(2,2)} = Pr{2 shows from d4}*Pr{2 shows from d3} = (1/4)*(1/3) = 1/12

Pr{(3,2)} = Pr{3 shows from d4}*Pr{2 shows from d3} = (1/4)*(1/3) = 1/12

Pr{(4,2)} = Pr{4 shows from d4}*Pr{2 shows from d3} = (1/4)*(1/3) = 1/12

Pr{(1,3)} = Pr{1 shows from d4}*Pr{3 shows from d3} = (1/4)*(1/3) = 1/12

Pr{(2,3)} = Pr{2 shows from d4}*Pr{3 shows from d3} = (1/4)*(1/3) = 1/12

Pr{(3,3)} = Pr{3 shows from d4}*Pr{3 shows from d3} = (1/4)*(1/3) = 1/12

Pr{(4,3)} = Pr{4 shows from d4}*Pr{3 shows from d3} = (1/4)*(1/3) = 1/12

Pair → Sum	1	2	3	4
1	(1,1) → 2	(2,1) → 3	(3,1) → 4	(4,1) → 5
2	(1,2) → 3	(2,2) → 4	(3,2) → 5	(4,2) → 6
3	(1,3) → 4	(2,3) → 5	(3,3) → 6	(4,3) → 7

Compute probabilities for each sum. Map the pairs to sums, and list the pairs that lead to each sum. Using the probabilities for each pair, compute probabilities for each sum value:

Pr{Sum=2} =

Pr{(1,1)} =

1/12

Pr{Sum=3} =

Pr{One of (1,2),(2,1) Shows} =

Pr{(1,2)}+Pr{(2,1)} =

(1/12)+(1/12)=

2/12

Pr{Sum=4} =

Pr{One of (1,3),(2,2),(3,1) Shows} =

Pr{(1,3)}+Pr{(2,2)}+Pr{(3,1)} =

(1/12)+(1/12) +(1/12)=

3/12

Pr{Sum=5} =

Pr{One of (4,1),(2,3),(3,2) Shows} =

Pr{(4,1)}+Pr{(2,3)}+Pr{(2,3)} =

(1/12)+(1/12) +(1/12)=

3/12

Pr{Sum=6} =

Pr{One of (4,2),(3,3) Shows} =

Pr{(4,2)}+Pr{(3,3)} =

(1/12)+(1/12)=

2/12

Pr{Sum=7} =

Pr{(4,3)} =

1/12

Case Study #1.8

Probability Computation Rules

Case Description: Compute selected probabilities associated with a pair of dice.

D4 model 

Face Value	Probability
1	4/10
2	3/10
3	2/10
4	1/10
Total	10/10

d3 model

Face Value	Probability
1	1/6
2	2/6
3	3/6
Total	1.00

The experiment: On each trial of the experiment, we toss the pair of dice (defined above) and observe the pair of faces that show.

Case Objectives: Lay out the possible face-pairs, and compute the probability for each pair. State any required assumptions.

Consider the random variable that maps the pair of face values into the sum of the face values.

The PAIR Model

There are 4*3=12 distinct pairs possible: Writing each pair as (d4 face value, d3 face value):

(1,1), (2,1), (3,1), (4,1), (1,2), (2,2), (3,2), (4,2), (1,3), (2,3), (3,3), (4,3)

(d4,d3)	1	2	3	4
1	(1,1)	(2,1)	(3,1)	(4,1)
2	(1,2)	(2,2)	(3,2)	(4,2)
3	(1,3)	(2,3)	(3,3)	(4,3)

Under the independent multiplication principle,

Pr{(event from d4, event from d3)} = Pr{ event from d4}*Pr{ event from d3}, so:

Pr{(1,1)} = Pr{1 shows from d4}*Pr{1 shows from d3} = (4/10)*(1/6) = 4/60

Pr{(2,1)} = Pr{2 shows from d4}*Pr{1 shows from d3} = (3/10)*(1/6) = 3/60

Pr{(3,1)} = Pr{3 shows from d4}*Pr{1 shows from d3} = (2/10)*(1/6) = 2/60

Pr{(4,1)} = Pr{4 shows from d4}*Pr{1 shows from d3} = (1/10)*(1/6) = 1/60

Pr{(1,2)} = Pr{1 shows from d4}*Pr{2 shows from d3} = (4/10)*(2/6) = 8/60

Pr{(2,2)} = Pr{2 shows from d4}*Pr{2 shows from d3} = (3/10)*(2/6) = 6/60

Pr{(3,2)} = Pr{3 shows from d4}*Pr{2 shows from d3} = (2/10)*(2/6) = 4/60

Pr{(4,2)} = Pr{4 shows from d4}*Pr{2 shows from d3} = (1/10)*(2/6) = 2/60

Pr{(1,3)} = Pr{1 shows from d4}*Pr{3 shows from d3} = (4/10)*(3/6) = 12/60

Pr{(2,3)} = Pr{2 shows from d4}*Pr{3 shows from d3} = (3/10)*(3/6) = 9/60

Pr{(3,3)} = Pr{3 shows from d4}*Pr{3 shows from d3} = (2/10)*(3/6) = 6/60

Pr{(4,3)} = Pr{4 shows from d4}*Pr{3 shows from d3} = (1/10)*(3/6) = 3/60

The SUM Model

Pair → Sum	1	2	3	4
1	(1,1) → 2	(2,1) → 3	(3,1) → 4	(4,1) → 5
2	(1,2) → 3	(2,2) → 4	(3,2) → 5	(4,2) → 6
3	(1,3) → 4	(2,3) → 5	(3,3) → 6	(4,3) → 7

Pr{Sum=2} = Pr{(1,1)} = 4/60

Pr{Sum=3} =

Pr{One of (1,2),(2,1) Shows} =

Pr{(1,2)}+Pr{(2,1)}=

(3/60)+(8/60)=

11/60

Pr{Sum=4} =

Pr{One of (3,1),(2,2),(1,3) Shows} =

Pr{(1,3)}+Pr{(2,2)}+Pr{(3,1)} =

(2/60)+(6/60)+(12/60)=

20/60

Pr{Sum=5} =

Pr{One of (4,1),(3,2),(2,3) Shows} =

Pr{(4,1)}+Pr{(3,2)}+Pr{(2,3)} =

(1/60)+(4/60) +(9/60)=

14/60

Pr{Sum=6} =

Pr{One of (4,2),(3,3) Shows} =

Pr{(4,2)}+Pr{(3,3)} = (2/60)+(6/60) =

8/60

Pr{Sum=7} =

Pr{(4,3)} =

3/60

Compare sample proportions (p) to model probabilities (P). Compare precision with increasing sample size.

Samples:

Fair Pair Model					Loaded Pair Model
Sample #1					Sample #5
Sum	n	p	P	E	Sum	n	p	P	E
2	11	0.055	0.083	16.667	2	11	0.055	0.067	13.400
3	33	0.165	0.167	33.333	3	35	0.175	0.183	36.600
4	51	0.255	0.250	50.000	4	75	0.375	0.333	66.600
5	60	0.300	0.250	50.000	5	49	0.245	0.233	46.600
6	30	0.150	0.167	33.333	6	25	0.125	0.133	26.600
7	15	0.075	0.083	16.667	7	5	0.025	0.050	10.000
Total	200	1	1	200	Total	200	1	1	200
Sample #2					Sample #6
Sum	n	p	P	E	Sum	n	p	P	E
2	13	0.065	0.083	16.667	2	13	0.065	0.067	13.4
3	43	0.215	0.167	33.333	3	34	0.17	0.183	36.6
4	57	0.285	0.250	50.000	4	77	0.385	0.333	66.6
5	49	0.245	0.250	50.000	5	40	0.2	0.233	46.6
6	23	0.115	0.167	33.333	6	27	0.135	0.133	26.6
7	15	0.075	0.083	16.667	7	9	0.045	0.05	10
Total	200	1	1	200	Total	200	1	1	200
Sample #3
Sum	n	p	P	E
2	25	0.125	0.083	16.667
3	28	0.140	0.167	33.333
4	56	0.280	0.250	50.000
5	48	0.240	0.250	50.000
6	37	0.185	0.167	33.333
7	6	0.030	0.083	16.667
Total	200	1	1	200
Pooled 123					Pooled 56
Sum	n	p	P	E	Sum	n	p	P	E
2	49	0.082	0.083	50.000	2	24	0.060	0.067	26.800
3	104	0.173	0.167	100.000	3	69	0.173	0.183	73.200
4	164	0.273	0.250	150.000	4	152	0.380	0.333	133.200
5	157	0.262	0.250	150.000	5	89	0.223	0.233	93.200
6	90	0.150	0.167	100.000	6	52	0.130	0.133	53.200
7	36	0.060	0.083	50.000	7	14	0.035	0.050	20.000
Total	600	1	1	600	Total	400	1	1	400

We’re seeing the pair model inheriting its probability structure from the individual dice. The random variable in turn inherits its probability structure from the pair model.

Rare Events

The Scale of Probability

Probabilities range from 0 to 1.

Pr{Event} = 0 The event is impossible.

Pr{Event} » 0 The event is rare, highly unlikely to be observed in modestly-sized samples.

Pr{Event} = ½ The event is observed in approximately half of observed trials. The event is as likely to occur as not.

Pr{Event} » 1 The event is almost certain, highly likely to be observed in nearly every observed trial.

Pr{Event} = 1 The event is certain.

Pr{Event} < ½ The event is more likely to not occur than to occur.

Pr{Event} > ½ The event is more likely to occur than not occur.

The Rare Event Approach

An event is rare if

Pr{Event} ≈ 0.

The implications for observing rare events in random samples are important. In particular, we can say that the smallest sample size in which we expect to reliably observe a rare event depends on its true probability. That is:

n ≥ 1/ Pr{Event}.

Rare Event Approach: Pairs of Dice and the Pair (1,1) Consider a sequence of pairs of fair dice, and the occurrence (relative to n=100) of the face-pair (1,1).

Pair of Fair Dice, each with face values {1,2,3} per die

(1^st D3, 2^nd D3)	1	2	3
1	(1,1)	(2,1)	(3,1)
2	(1,2)	(2,2)	(3,2)
3	(1,3)	(2,3)	(3,3)

Pr{(1,1) shows} = Pr{1 shows from 1^st D3}*Pr{1 shows from 2^nd D3} = (1/3)*(1/3) = 1/9 @ .1111111

In random samples of 100 tosses of the pair of dice, we expect approximately 100*Pr{(1,1)} = 100*(1/9) @ 11.11111.

The smallest sample size for which we expect to observe one or more tosses showing the pair (1,1) is 1/(1/9) = 9.

Pair of Fair Dice, one with face values {1,2,3,4} per die and one with face values {1,2,3} per die

(1^st D4, 2^nd D3)	1	2	3	4
1	(1,1)	(2,1)	(3,1)	(4,1)
2	(1,2)	(2,2)	(3,2)	(4,2)
3	(1,3)	(2,3)	(3,3)	(4,3)

Pr{(1,1) shows} = Pr{1 shows from 1^st D4}*Pr{1 shows from 2^nd D3} = (1/4)*(1/3) = 1/12 @ .0833

In random samples of 100 tosses of the pair of dice, we expect approximately 100*Pr{(1,1)} = 100*(1/12) @ 8.33

The smallest sample size for which we expect to observe one or more tosses showing the pair (1,1) is 1/(1/12) = 12.

Pair of Fair Dice, each with face values {1,2,3,4}

(1^st D4, 2^nd D4)	1	2	3	4
1	(1,1)	(2,1)	(3,1)	(4,1)
2	(1,2)	(2,2)	(3,2)	(4,2)
3	(1,3)	(2,3)	(3,3)	(4,3)
4	(1,4)	(2,4)	(3,4)	(4,4)

Pr{(1,1) shows} = Pr{1 shows from 1^st D4}*Pr{1 shows from 2^nd D4} = (1/4)*(1/4) = 1/16 @ .0625

In random samples of 100 tosses of the pair of dice, we expect approximately 100*Pr{(1,1)} = 100*(1/16) @ 6.25

The smallest sample size for which we expect to observe one or more tosses showing the pair (1,1) is 1/(1/16) = 16.

Pair of Fair Dice, one with face values {1,2,3,4,5} and one with face values {1,2,3,4}

(1^st D5, 2^nd D4)	1	2	3	4	5
1	(1,1)	(2,1)	(3,1)	(4,1)	(5,1)
2	(1,2)	(2,2)	(3,2)	(4,2)	(5,2)
3	(1,3)	(2,3)	(3,3)	(4,3)	(5,3)
4	(1,4)	(2,4)	(3,4)	(4,4)	(5,4)

Pr{(1,1) shows} = Pr{1 shows from 1^st D5}*Pr{1 shows from 2^nd D4} = (1/5)*(1/4) = 1/20 @ .05

In random samples of 100 tosses of the pair of dice, we expect approximately 100*Pr{(1,1)} = 100*(1/20) @ 5

The smallest sample size for which we expect to observe one or more tosses showing the pair (1,1) is 1/(1/20) = 20.

Pair of Fair Dice, each with face values {1,2,3,4,5}

(1^st D5, 2^nd D5)	1	2	3	4	5
1	(1,1)	(2,1)	(3,1)	(4,1)	(5,1)
2	(1,2)	(2,2)	(3,2)	(4,2)	(5,2)
3	(1,3)	(2,3)	(3,3)	(4,3)	(5,3)
4	(1,4)	(2,4)	(3,4)	(4,4)	(5,4)
5	(1,5)	(2,5)	(3,5)	(4,5)	(5,5)

Pr{(1,1) shows} = Pr{1 shows from 1^st D5}*Pr{1 shows from 2^nd D5} = (1/5)*(1/5) = 1/25 @ .04

In random samples of 100 tosses of the pair of dice, we expect approximately 100*Pr{(1,1)} = 100*(1/25) @ 4

The smallest sample size for which we expect to observe one or more tosses showing the pair (1,1) is 1/(1/25) = 25.

Pair of Fair Dice, one with face values {1,2,3,4,5,6} and one with face values {1,2,3,4,5}

(1^st D6, 2^nd D5)	1	2	3	4	5	6
1	(1,1)	(2,1)	(3,1)	(4,1)	(5,1)	(6,1)
2	(1,2)	(2,2)	(3,2)	(4,2)	(5,2)	(6,2)
3	(1,3)	(2,3)	(3,3)	(4,3)	(5,3)	(6,3)
4	(1,4)	(2,4)	(3,4)	(4,4)	(5,4)	(6,4)
5	(1,5)	(2,5)	(3,5)	(4,5)	(5,5)	(6,5)

Pr{(1,1) shows} = Pr{1 shows from 1^st D6}*Pr{1 shows from 2^nd D5} = (1/6)*(1/5) = 1/30 @ .0333

In random samples of 100 tosses of the pair of dice, we expect approximately 100*Pr{(1,1)} = 100*(1/30) @ 3.33

The smallest sample size for which we expect to observe one or more tosses showing the pair (1,1) is 1/(1/30) = 30.

Pair of Fair Dice, each with face values {1,2,3,4,5,6}

(1^st D6, 2^nd D6)	1	2	3	4	5	6
1	(1,1)	(2,1)	(3,1)	(4,1)	(5,1)	(6,1)
2	(1,2)	(2,2)	(3,2)	(4,2)	(5,2)	(6,2)
3	(1,3)	(2,3)	(3,3)	(4,3)	(5,3)	(6,3)
4	(1,4)	(2,4)	(3,4)	(4,4)	(5,4)	(6,4)
5	(1,5)	(2,5)	(3,5)	(4,5)	(5,5)	(6,5)
6	(1,6)	(2,6)	(3,6)	(4,6)	(5,6)	(6,6)

Pr{(1,1) shows} = Pr{1 shows from 1^st D6}*Pr{1 shows from 2^nd D6} = (1/6)*(1/6) = 1/36 @ .0278

In random samples of 100 tosses of the pair of dice, we expect approximately 100*Pr{(1,1)} = 100*(1/36) @ 2.78

The smallest sample size for which we expect to observe one or more tosses showing the pair (1,1) is 1/(1/36) = 36.

Pair of Fair Dice, one with face values {1,2,3,4,5,6,7,8} and one with face values {1,2,3,4,5,6}

(1^st D8, 2^nd D6)	1	2	3	4	5	6	7	8
1	(1,1)	(2,1)	(3,1)	(4,1)	(5,1)	(6,1)	(7,1)	(8,1)
2	(1,2)	(2,2)	(3,2)	(4,2)	(5,2)	(6,2)	(7,2)	(8,2)
3	(1,3)	(2,3)	(3,3)	(4,3)	(5,3)	(6,3)	(7,3)	(8,3)
4	(1,4)	(2,4)	(3,4)	(4,4)	(5,4)	(6,4)	(7,4)	(8,4)
5	(1,5)	(2,5)	(3,5)	(4,5)	(5,5)	(6,5)	(7,5)	(8,5)
6	(1,6)	(2,6)	(3,6)	(4,6)	(5,6)	(6,6)	(7,6)	(8,6)

Pr{(1,1) shows} = Pr{1 shows from 1^st D8}*Pr{1 shows from 2^nd D6} = (1/8)*(1/6) = 1/48 @ .0208

In random samples of 100 tosses of the pair of dice, we expect approximately 100*Pr{(1,1)} = 100*(1/48) @ 2.08

The smallest sample size for which we expect to observe one or more tosses showing the pair (1,1) is 1/(1/48) = 48.

Pair of Fair Dice, each with face values {1,2,3,4,5,6,7,8}

(1^st D8, 2^nd D8)	1	2	3	4	5	6	7	8
1	(1,1)	(2,1)	(3,1)	(4,1)	(5,1)	(6,1)	(7,1)	(8,1)
2	(1,2)	(2,2)	(3,2)	(4,2)	(5,2)	(6,2)	(7,2)	(8,2)
3	(1,3)	(2,3)	(3,3)	(4,3)	(5,3)	(6,3)	(7,3)	(8,3)
4	(1,4)	(2,4)	(3,4)	(4,4)	(5,4)	(6,4)	(7,4)	(8,4)
5	(1,5)	(2,5)	(3,5)	(4,5)	(5,5)	(6,5)	(7,5)	(8,5)
6	(1,6)	(2,6)	(3,6)	(4,6)	(5,6)	(6,6)	(7,6)	(8,6)
7	(1,7)	(2,7)	(3,7)	(4,7)	(5,7)	(6,7)	(7,7)	(8,7)
8	(1,8)	(2,8)	(3,8)	(4,8)	(5,8)	(6,8)	(7,8)	(8,8)

Pr{(1,1) shows} = Pr{1 shows from 1^st D8}*Pr{1 shows from 2^nd D8} = (1/8)*(1/8) = 1/64 @ 0.0156

In random samples of 100 tosses of the pair of dice, we expect approximately 100*Pr{(1,1)} = 100*(1/64) @ 1.56.

The smallest sample size for which we expect to observe one or more tosses showing the pair (1,1) is 1/(1/64) = 64.

Pair of Fair Dice, on e with face values {1,2,3,4,5,6,7,8,9,10} and one with face values {1,2,3,4,5,6,7,8}

(1^st D10, 2^nd D8)	1	2	3	4	5	6	7	8	9	10
1	(1,1)	(2,1)	(3,1)	(4,1)	(5,1)	(6,1)	(7,1)	(8,1)	(9,1)	(10,1)
2	(1,2)	(2,2)	(3,2)	(4,2)	(5,2)	(6,2)	(7,2)	(8,2)	(9,2)	(10,2)
3	(1,3)	(2,3)	(3,3)	(4,3)	(5,3)	(6,3)	(7,3)	(8,3)	(9,3)	(10,3)
4	(1,4)	(2,4)	(3,4)	(4,4)	(5,4)	(6,4)	(7,4)	(8,4)	(9,4)	(10,4)
5	(1,5)	(2,5)	(3,5)	(4,5)	(5,5)	(6,5)	(7,5)	(8,5)	(9,5)	(10,5)
6	(1,6)	(2,6)	(3,6)	(4,6)	(5,6)	(6,6)	(7,6)	(8,6)	(9,6)	(10,6)
7	(1,7)	(2,7)	(3,7)	(4,7)	(5,7)	(6,7)	(71,7)	(8,7)	(9,7)	(10,7)
8	(1,8)	(2,8)	(3,8)	(4,8)	(5,8)	(6,8)	(7,8)	(8,8)	(9,8)	(10,8)

Pr{(1,1) shows} = Pr{1 shows from 1^st D10}*Pr{1 shows from 2^nd D8} = (1/10)*(1/8) = 1/80 = 0.0125

In random samples of 100 tosses of the pair of dice, we expect approximately 100*Pr{(1,1)} = 100*(1/80) = 1.25.

The smallest sample size for which we expect to observe one or more tosses showing the pair (1,1) is 1/(1/80) = 80.

Pair of Fair Dice, each with face values {1,2,3,4,5,6,7,8,9,10}

(1st D10, 2nd D10)	1	2	3	4	5	6	7	8	9	10
1	(1,1)	(2,1)	(3,1)	(4,1)	(5,1)	(6,1)	(7,1)	(8,1)	(9,1)	(10,1)
2	(1,2)	(2,2)	(3,2)	(4,2)	(5,2)	(6,2)	(7,2)	(8,2)	(9,2)	(10,2)
3	(1,3)	(2,3)	(3,3)	(4,3)	(5,3)	(6,3)	(7,3)	(8,3)	(9,3)	(10,3)
4	(1,4)	(2,4)	(3,4)	(4,4)	(5,4)	(6,4)	(7,4)	(8,4)	(9,4)	(10,4)
5	(1,5)	(2,5)	(3,5)	(4,5)	(5,5)	(6,5)	(7,5)	(8,5)	(9,5)	(10,5)
6	(1,6)	(2,6)	(3,6)	(4,6)	(5,6)	(6,6)	(7,6)	(8,6)	(9,6)	(10,6)
7	(1,7)	(2,7)	(3,7)	(4,7)	(5,7)	(6,7)	(71,7)	(8,7)	(9,7)	(10,7)
8	(1,8)	(2,8)	(3,8)	(4,8)	(5,8)	(6,8)	(7,8)	(8,8)	(9,8)	(10,8)
9	(1,9)	(2,9)	(3,9)	(4,9)	(5,9)	(6,9)	(7,9)	(8,9)	(9,9)	(10,9)
10	(1,10)	(2,10)	(3,10)	(4,10)	(5,10)	(6,10)	(7,10)	(8,10)	(9,10)	(10,10)

Pr{(1,1) shows} = Pr{1 shows from 1^st D10}*Pr{1 shows from 2^nd D10} = (1/10)*(1/10) = 1/100 = 0.01

In random samples of 100 tosses of the pair of dice, we expect approximately 100*Pr{(1,1)} = 100*(1/100) = 1.

The smallest sample size for which we expect to observe one or more tosses showing the pair (1,1) is 1/(1/100) = 100.

Pair of Fair Dice, one with face values {1,2,3,4,5,6,7,8,9,10,11,12} and one with face values {1,2,3,4,5,6,7,8}

(1st D12, 2nd D8)	1	2	3	4	5	6	7	8	9	10	11	12
1	(1,1)	(2,1)	(3,1)	(4,1)	(5,1)	(6,1)	(7,1)	(8,1)	(9,1)	(10,1)	(11,1)	(12,1)
2	(1,2)	(2,2)	(3,2)	(4,2)	(5,2)	(6,2)	(7,2)	(8,2)	(9,2)	(10,2)	(11,2)	(12,2)
3	(1,3)	(2,3)	(3,3)	(4,3)	(5,3)	(6,3)	(7,3)	(8,3)	(9,3)	(10,3)	(11,3)	(12,3)
4	(1,4)	(2,4)	(3,4)	(4,4)	(5,4)	(6,4)	(7,4)	(8,4)	(9,4)	(10,4)	(11,4)	(12,4)
5	(1,5)	(2,5)	(3,5)	(4,5)	(5,5)	(6,5)	(7,5)	(8,5)	(9,5)	(10,5)	(11,5)	(12,5)
6	(1,6)	(2,6)	(3,6)	(4,6)	(5,6)	(6,6)	(7,6)	(8,6)	(9,6)	(10,6)	(11,6)	(12,6)
7	(1,7)	(2,7)	(3,7)	(4,7)	(5,7)	(6,7)	(71,7)	(8,7)	(9,7)	(10,7)	(11,7)	(12,7)
8	(1,8)	(2,8)	(3,8)	(4,8)	(5,8)	(6,8)	(7,8)	(8,8)	(9,8)	(10,8)	(11,8)	(12,8)

Pr{(1,1) shows} = Pr{1 shows from D12}*Pr{1 shows from D8} = (1/12)*(1/8) = 1/96 @ 0.0104

In random samples of 100 tosses of the pair of dice, we expect approximately 100*Pr{(1,1)} = 100*(1/96) @ 1.04.

The smallest sample size for which we expect to observe one or more tosses showing the pair (1,1) is 1/(1/96) = 96.

Pair of Fair Dice, one with face values {1,2,3,4,5,6,7,8,9,10,11,12} and one with face values {1,2,3,4,5,6,7,8,9,10}

(1,1)	(2,1)	(3,1)	(4,1)	(5,1)	(6,1)	(7,1)	(8,1)	(9,1)	(10,1)	(11,1)	(12,1)
(1,2)	(2,2)	(3,2)	(4,2)	(5,2)	(6,2)	(7,2)	(8,2)	(9,2)	(10,2)	(11,2)	(12,2)
(1,3)	(2,3)	(3,3)	(4,3)	(5,3)	(6,3)	(7,3)	(8,3)	(9,3)	(10,3)	(11,3)	(12,3)
(1,4)	(2,4)	(3,4)	(4,4)	(5,4)	(6,4)	(7,4)	(8,4)	(9,4)	(10,4)	(11,4)	(12,4)
(1,5)	(2,5)	(3,5)	(4,5)	(5,5)	(6,5)	(7,5)	(8,5)	(9,5)	(10,5)	(11,5)	(12,5)
(1,6)	(2,6)	(3,6)	(4,6)	(5,6)	(6,6)	(7,6)	(8,6)	(9,6)	(10,6)	(11,6)	(12,6)
(1,7)	(2,7)	(3,7)	(4,7)	(5,7)	(6,7)	(71,7)	(8,7)	(9,7)	(10,7)	(11,7)	(12,7)
(1,8)	(2,8)	(3,8)	(4,8)	(5,8)	(6,8)	(7,8)	(8,8)	(9,8)	(10,8)	(11,8)	(12,8)
(1,9)	(2,9)	(3,9)	(4,9)	(5,9)	(6,9)	(7,9)	(8,9)	(9,9)	(10,9)	(11,9)	(12,9)
(1,10)	(2,10)	(3,10)	(4,10)	(5,10)	(6,10)	(7,10)	(8,10)	(9,10)	(10,10)	(11,10)	(12,10)

Pr{(1,1) shows} = Pr{1 shows from D12}*Pr{1 shows from D10} = (1/12)*(1/10) = 1/120 @ 0.00833

In random samples of 100 tosses of the pair of dice, we expect approximately 100*Pr{(1,1)} = 100*(1/120) @ 0.833.

The smallest sample size for which we expect to observe one or more tosses showing the pair (1,1) is 1/(1/120) = 120.

Pair of Fair Dice, each with face values {1,2,3,4,5,6,7,8,9,10,11,12}

(1st D12, 2nd D12)	1	2	3	4	5	6	7	8	9	10	11	12
1	(1,1)	(2,1)	(3,1)	(4,1)	(5,1)	(6,1)	(7,1)	(8,1)	(9,1)	(10,1)	(11,1)	(12,1)
2	(1,2)	(2,2)	(3,2)	(4,2)	(5,2)	(6,2)	(7,2)	(8,2)	(9,2)	(10,2)	(11,2)	(12,2)
3	(1,3)	(2,3)	(3,3)	(4,3)	(5,3)	(6,3)	(7,3)	(8,3)	(9,3)	(10,3)	(11,3)	(12,3)
4	(1,4)	(2,4)	(3,4)	(4,4)	(5,4)	(6,4)	(7,4)	(8,4)	(9,4)	(10,4)	(11,4)	(12,4)
5	(1,5)	(2,5)	(3,5)	(4,5)	(5,5)	(6,5)	(7,5)	(8,5)	(9,5)	(10,5)	(11,5)	(12,5)
6	(1,6)	(2,6)	(3,6)	(4,6)	(5,6)	(6,6)	(7,6)	(8,6)	(9,6)	(10,6)	(11,6)	(12,6)
7	(1,7)	(2,7)	(3,7)	(4,7)	(5,7)	(6,7)	(71,7)	(8,7)	(9,7)	(10,7)	(11,7)	(12,7)
8	(1,8)	(2,8)	(3,8)	(4,8)	(5,8)	(6,8)	(7,8)	(8,8)	(9,8)	(10,8)	(11,8)	(12,8)
9	(1,9)	(2,9)	(3,9)	(4,9)	(5,9)	(6,9)	(7,9)	(8,9)	(9,9)	(10,9)	(11,9)	(12,9)
10	(1,10)	(2,10)	(3,10)	(4,10)	(5,10)	(6,10)	(7,10)	(8,10)	(9,10)	(10,10)	(11,10)	(12,10)
11	(1,11)	(2,11)	(3,11)	(4,11)	(5,11)	(6,11)	(7,11)	(8,11)	(9,11)	(10,11)	(11,11)	(12,11)
12	(1,12)	(2,12)	(3,12)	(4,12)	(5,12)	(6,12)	(7,12)	(8,12)	(9,12)	(10,12)	(11,12)	(12,12)

Pr{(1,1) shows} = Pr{1 shows from 1^st D12}*Pr{1 shows from 2^nd D12} = (1/12)*(1/12) = 1/144 @ 0.006944444

In random samples of 100 tosses of the pair of dice, we expect approximately 100*Pr{(1,1)} = 100*(1/144) @ . 6944444

The smallest sample size for which we expect to observe one or more tosses showing the pair (1,1) is 1/(1/144) = 144.

Pair of Fair Dice, one with face values {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20} and one with face values {1,2,3,4,5,6,7,8,9,10}

(1,1)	(2,1)	(3,1)	(4,1)	(5,1)	(6,1)	(7,1)	(8,1)	(9,1)	(10,1)	(11,1)	(12,1)	(13,1)	(14,1)	(15,1)	(16,1)	(17,1)	(18,1)	(19,1)	(20,1)
(1,2)	(2,2)	(3,2)	(4,2)	(5,2)	(6,2)	(7,2)	(8,2)	(9,2)	(10,2)	(11,2)	(12,2)	(13,2)	(14,2)	(15,2)	(16,2)	(17,2)	(18,2)	(19,2)	(20,2)
(1,3)	(2,3)	(3,3)	(4,3)	(5,3)	(6,3)	(7,3)	(8,3)	(9,3)	(10,3)	(11,3)	(12,3)	(13,3)	(14,3)	(15,3)	(16,3)	(17,3)	(18,3)	(19,3)	(20,3)
(1,4)	(2,4)	(3,4)	(4,4)	(5,4)	(6,4)	(7,4)	(8,4)	(9,4)	(10,4)	(11,4)	(12,4)	(13,4)	(14,4)	(15,4)	(16,4)	(17,4)	(18,4)	(19,4)	(20,4)
(1,5)	(2,5)	(3,5)	(4,5)	(5,5)	(6,5)	(7,5)	(8,5)	(9,5)	(10,5)	(11,5)	(12,5)	(13,5)	(14,5)	(15,5)	(16,5)	(17,5)	(18,5)	(19,5)	(20,5)
(1,6)	(2,6)	(3,6)	(4,6)	(5,6)	(6,6)	(7,6)	(8,6)	(9,6)	(10,6)	(11,6)	(12,6)	(13,6)	(14,6)	(15,6)	(16,6)	(17,6)	(18,6)	(19,6)	(20,6)
(1,7)	(2,7)	(3,7)	(4,7)	(5,7)	(6,7)	(71,7)	(8,7)	(9,7)	(10,7)	(11,7)	(12,7)	(13,7)	(14,7)	(15,7)	(16,7)	(17,7)	(18,7)	(19,7)	(20,7)
(1,8)	(2,8)	(3,8)	(4,8)	(5,8)	(6,8)	(7,8)	(8,8)	(9,8)	(10,8)	(11,8)	(12,8)	(13,8)	(14,8)	(15,8)	(16,8)	(17,8)	(18,8)	(19,8)	(20,8)
(1,9)	(2,9)	(3,9)	(4,9)	(5,9)	(6,9)	(7,9)	(8,9)	(9,9)	(10,9)	(11,9)	(12,9)	(13,9)	(14,9)	(15,9)	(16,9)	(17,9)	(18,9)	(19,9)	(20,9)
(1,10)	(2,10)	(3,10)	(4,10)	(5,10)	(6,10)	(7,10)	(8,10)	(9,10)	(10,10)	(11,10)	(12,10)	(13,10)	(14,10)	(15,10)	(16,10)	(17,10)	(18,10)	(19,10)	(20,10)

Pr{(1,1) shows} = Pr{1 shows from 1^st D20}*Pr{1 shows from 2^nd D10} = (1/20)*(1/10) = 1/200 @ 0.005

In random samples of 100 tosses of the pair of dice, we expect approximately 100*Pr{(1,1)} = 100*(1/200) @ .50

The smallest sample size for which we expect to observe one or more tosses showing the pair (1,1) is 1/(1/200) = 200.

Pair of Fair Dice, one with face values {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24} and one with face values {1,2,3,4,5,6,7,8,9,10}

(1,1)	(2,1)	(3,1)	(4,1)	(5,1)	(6,1)	(7,1)	(8,1)	(9,1)	(10,1)	(11,1)	(12,1)
(1,2)	(2,2)	(3,2)	(4,2)	(5,2)	(6,2)	(7,2)	(8,2)	(9,2)	(10,2)	(11,2)	(12,2)
(1,3)	(2,3)	(3,3)	(4,3)	(5,3)	(6,3)	(7,3)	(8,3)	(9,3)	(10,3)	(11,3)	(12,3)
(1,4)	(2,4)	(3,4)	(4,4)	(5,4)	(6,4)	(7,4)	(8,4)	(9,4)	(10,4)	(11,4)	(12,4)
(1,5)	(2,5)	(3,5)	(4,5)	(5,5)	(6,5)	(7,5)	(8,5)	(9,5)	(10,5)	(11,5)	(12,5)
(1,6)	(2,6)	(3,6)	(4,6)	(5,6)	(6,6)	(7,6)	(8,6)	(9,6)	(10,6)	(11,6)	(12,6)
(1,7)	(2,7)	(3,7)	(4,7)	(5,7)	(6,7)	(71,7)	(8,7)	(9,7)	(10,7)	(11,7)	(12,7)
(1,8)	(2,8)	(3,8)	(4,8)	(5,8)	(6,8)	(7,8)	(8,8)	(9,8)	(10,8)	(11,8)	(12,8)
(1,9)	(2,9)	(3,9)	(4,9)	(5,9)	(6,9)	(7,9)	(8,9)	(9,9)	(10,9)	(11,9)	(12,9)
(1,10)	(2,10)	(3,10)	(4,10)	(5,10)	(6,10)	(7,10)	(8,10)	(9,10)	(10,10)	(11,10)	(12,10)
(13,1)	(14,1)	(15,1)	(16,1)	(17,1)	(18,1)	(19,1)	(20,1)	(21,1)	(22,1)	(23,1)	(24,1)
(13,2)	(14,2)	(15,2)	(16,2)	(17,2)	(18,2)	(19,2)	(20,2)	(21,2)	(22,2)	(23,2)	(24,2)
(13,3)	(14,3)	(15,3)	(16,3)	(17,3)	(18,3)	(19,3)	(20,3)	(21,3)	(22,3)	(23,3)	(24,3)
(13,4)	(14,4)	(15,4)	(16,4)	(17,4)	(18,4)	(19,4)	(20,4)	(21,4)	(22,4)	(23,4)	(24,4)
(13,5)	(14,5)	(15,5)	(16,5)	(17,5)	(18,5)	(19,5)	(20,5)	(21,5)	(22,5)	(23,5)	(24,5)
(13,6)	(14,6)	(15,6)	(16,6)	(17,6)	(18,6)	(19,6)	(20,6)	(21,6)	(22,6)	(23,6)	(24,6)
(13,7)	(14,7)	(15,7)	(16,7)	(17,7)	(18,7)	(19,7)	(20,7)	(21,7)	(22,7)	(23,7)	(24,7)
(13,8)	(14,8)	(15,8)	(16,8)	(17,8)	(18,8)	(19,8)	(20,8)	(21,8)	(22,8)	(23,8)	(24,8)
(13,9)	(14,9)	(15,9)	(16,9)	(17,9)	(18,9)	(19,9)	(20,9)	(21,9)	(22,9)	(23,9)	(24,9)
(13,10)	(14,10)	(15,10)	(16,10)	(17,10)	(18,10)	(19,10)	(20,10)	(21,10)	(22,10)	(23,10)	(24,10)

Pr{(1,1) shows} = Pr{1 shows from 1^st D24}*Pr{1 shows from 2^nd D10} = (1/24)*(1/10) = 1/240 @ 0.0042

In random samples of 100 tosses of the pair of dice, we expect approximately 100*Pr{(1,1)} = 100*(1/240) @ .42

The smallest sample size for which we expect to observe one or more tosses showing the pair (1,1) is 1/(1/240) = 240.

Pair of Fair Dice, one with face values {1,2,3,4,5,6,7,8,9,10,11,27,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30} and one with face values {1,2,3,4,5,6,7,8,9,10}

(1,1)	(2,1)	(3,1)	(4,1)	(5,1)	(6,1)	(7,1)	(8,1)	(9,1)	(10,1)	(11,1)	(12,1)	(13,1)	(14,1)	(15,1)
(1,2)	(2,2)	(3,2)	(4,2)	(5,2)	(6,2)	(7,2)	(8,2)	(9,2)	(10,2)	(11,2)	(12,2)	(13,2)	(14,2)	(15,2)
(1,3)	(2,3)	(3,3)	(4,3)	(5,3)	(6,3)	(7,3)	(8,3)	(9,3)	(10,3)	(11,3)	(12,3)	(13,3)	(14,3)	(15,3)
(1,4)	(2,4)	(3,4)	(4,4)	(5,4)	(6,4)	(7,4)	(8,4)	(9,4)	(10,4)	(11,4)	(12,4)	(13,4)	(14,4)	(15,4)
(1,5)	(2,5)	(3,5)	(4,5)	(5,5)	(6,5)	(7,5)	(8,5)	(9,5)	(10,5)	(11,5)	(12,5)	(13,5)	(14,5)	(15,5)
(1,6)	(2,6)	(3,6)	(4,6)	(5,6)	(6,6)	(7,6)	(8,6)	(9,6)	(10,6)	(11,6)	(12,6)	(13,6)	(14,6)	(15,6)
(1,7)	(2,7)	(3,7)	(4,7)	(5,7)	(6,7)	(71,7)	(8,7)	(9,7)	(10,7)	(11,7)	(12,7)	(13,7)	(14,7)	(15,7)
(1,8)	(2,8)	(3,8)	(4,8)	(5,8)	(6,8)	(7,8)	(8,8)	(9,8)	(10,8)	(11,8)	(12,8)	(13,8)	(14,8)	(15,8)
(1,9)	(2,9)	(3,9)	(4,9)	(5,9)	(6,9)	(7,9)	(8,9)	(9,9)	(10,9)	(11,9)	(12,9)	(13,9)	(14,9)	(15,9)
(1,10)	(2,10)	(3,10)	(4,10)	(5,10)	(6,10)	(7,10)	(8,10)	(9,10)	(10,10)	(11,10)	(12,10)	(13,10)	(14,10)	(15,10)
(16,1)	(17,1)	(18,1)	(19,1)	(20,1)	(21,1)	(22,1)	(23,1)	(24,1)	(25,1)	(26,1)	(27,1)	(28,1)	(29,1)	(30,1)
(16,2)	(17,2)	(18,2)	(19,2)	(20,2)	(21,2)	(22,2)	(23,2)	(24,2)	(25,2)	(26,2)	(27,2)	(28,2)	(29,2)	(30,2)
(16,3)	(17,3)	(18,3)	(19,3)	(20,3)	(21,3)	(22,3)	(23,3)	(24,3)	(25,3)	(26,3)	(27,3)	(28,3)	(29,3)	(30,3)
(16,4)	(17,4)	(18,4)	(19,4)	(20,4)	(21,4)	(22,4)	(23,4)	(24,4)	(25,4)	(26,4)	(27,4)	(28,4)	(29,4)	(30,4)
(16,5)	(17,5)	(18,5)	(19,5)	(20,5)	(21,5)	(22,5)	(23,5)	(24,5)	(25,5)	(26,5)	(27,5)	(28,5)	(29,5)	(30,5)
(16,6)	(17,6)	(18,6)	(19,6)	(20,6)	(21,6)	(22,6)	(23,6)	(24,6)	(25,6)	(26,6)	(27,6)	(28,6)	(29,6)	(30,6)
(16,7)	(17,7)	(18,7)	(19,7)	(20,7)	(21,7)	(71,7)	(23,7)	(24,7)	(25,7)	(26,7)	(27,7)	(28,7)	(29,7)	(30,7)
(16,8)	(17,8)	(18,8)	(19,8)	(20,8)	(21,8)	(22,8)	(23,8)	(24,8)	(25,8)	(26,8)	(27,8)	(28,8)	(29,8)	(30,8)
(16,9)	(17,9)	(18,9)	(19,9)	(20,9)	(21,9)	(22,9)	(23,9)	(24,9)	(25,9)	(26,9)	(27,9)	(28,9)	(29,9)	(30,9)
(16,10)	(17,10)	(18,10)	(19,10)	(20,10)	(21,10)	(22,10)	(23,10)	(24,10)	(25,10)	(26,10)	(27,10)	(28,10)	(29,10)	(30,10)

Pr{(1,1) shows} = Pr{1 shows from 1^st D30}*Pr{1 shows from 2^nd D10} = (1/30)*(1/10) = 1/300 @ 0.003333

In random samples of 100 tosses of the pair of dice, we expect approximately 100*Pr{(1,1)} = 100*(1/300) @ 0.03333

The smallest sample size for which we expect to observe one or more tosses showing the pair (1,1) is 1/(1/300) = 300.

Pair of Fair Dice, each with face values {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}

(1,1)	(2,1)	(3,1)	(4,1)	(5,1)	(6,1)	(7,1)	(8,1)	(9,1)	(10,1)	(11,1)	(12,1)	(13,1)	(14,1)	(15,1)	(16,1)	(17,1)	(18,1)	(19,1)	(20,1)
(1,2)	(2,2)	(3,2)	(4,2)	(5,2)	(6,2)	(7,2)	(8,2)	(9,2)	(10,2)	(11,2)	(12,2)	(13,2)	(14,2)	(15,2)	(16,2)	(17,2)	(18,2)	(19,2)	(20,2)
(1,3)	(2,3)	(3,3)	(4,3)	(5,3)	(6,3)	(7,3)	(8,3)	(9,3)	(10,3)	(11,3)	(12,3)	(13,3)	(14,3)	(15,3)	(16,3)	(17,3)	(18,3)	(19,3)	(20,3)
(1,4)	(2,4)	(3,4)	(4,4)	(5,4)	(6,4)	(7,4)	(8,4)	(9,4)	(10,4)	(11,4)	(12,4)	(13,4)	(14,4)	(15,4)	(16,4)	(17,4)	(18,4)	(19,4)	(20,4)
(1,5)	(2,5)	(3,5)	(4,5)	(5,5)	(6,5)	(7,5)	(8,5)	(9,5)	(10,5)	(11,5)	(12,5)	(13,5)	(14,5)	(15,5)	(16,5)	(17,5)	(18,5)	(19,5)	(20,5)
(1,6)	(2,6)	(3,6)	(4,6)	(5,6)	(6,6)	(7,6)	(8,6)	(9,6)	(10,6)	(11,6)	(12,6)	(13,6)	(14,6)	(15,6)	(16,6)	(17,6)	(18,6)	(19,6)	(20,6)
(1,7)	(2,7)	(3,7)	(4,7)	(5,7)	(6,7)	(71,7)	(8,7)	(9,7)	(10,7)	(11,7)	(12,7)	(13,7)	(14,7)	(15,7)	(16,7)	(17,7)	(18,7)	(19,7)	(20,7)
(1,8)	(2,8)	(3,8)	(4,8)	(5,8)	(6,8)	(7,8)	(8,8)	(9,8)	(10,8)	(11,8)	(12,8)	(13,8)	(14,8)	(15,8)	(16,8)	(17,8)	(18,8)	(19,8)	(20,8)
(1,9)	(2,9)	(3,9)	(4,9)	(5,9)	(6,9)	(7,9)	(8,9)	(9,9)	(10,9)	(11,9)	(12,9)	(13,9)	(14,9)	(15,9)	(16,9)	(17,9)	(18,9)	(19,9)	(20,9)
(1,10)	(2,10)	(3,10)	(4,10)	(5,10)	(6,10)	(7,10)	(8,10)	(9,10)	(10,10)	(11,10)	(12,10)	(13,10)	(14,10)	(15,10)	(16,10)	(17,10)	(18,10)	(19,10)	(20,10)
(1,11)	(2,11)	(3,11)	(4,11)	(5,11)	(6,11)	(7,11)	(8,11)	(9,11)	(10,11)	(11,11)	(12,11)	(13,11)	(14,11)	(15,11)	(16,11)	(17,11)	(18,11)	(19,11)	(20,11)
(1,12)	(2,12)	(3,12)	(4,12)	(5,12)	(6,12)	(7,12)	(8,12)	(9,12)	(10,12)	(11,12)	(12,12)	(13,12)	(14,12)	(15,12)	(16,12)	(17,12)	(18,12)	(19,12)	(20,12)
(1,13)	(2,13)	(3,13)	(4,13)	(5,13)	(6,13)	(7,13)	(8,13)	(9,13)	(10,13)	(11,13)	(12,13)	(13,13)	(14,13)	(15,13)	(16,13)	(17,13)	(18,13)	(19,13)	(20,13)
(1,14)	(2,14)	(3,14)	(4,14)	(5,14)	(6,14)	(7,14)	(8,14)	(9,14)	(10,14)	(11,14)	(12,14)	(13,14)	(14,14)	(15,14)	(16,14)	(17,14)	(18,14)	(19,14)	(20,14)
(1,15)	(2,15)	(3,15)	(4,15)	(5,15)	(6,15)	(7,15)	(8,15)	(9,15)	(10,15)	(11,15)	(12,15)	(13,15)	(14,15)	(15,15)	(16,15)	(17,15)	(18,15)	(19,15)	(20,15)
(1,16)	(2,16)	(3,16)	(4,16)	(5,16)	(6,16)	(7,16)	(8,16)	(9,16)	(10,16)	(11,16)	(12,16)	(13,16)	(14,16)	(15,16)	(16,16)	(17,16)	(18,16)	(19,16)	(20,16)
(1,17)	(2,17)	(3,17)	(4,17)	(5,17)	(6,17)	(71,17)	(8,17)	(9,17)	(10,17)	(11,17)	(12,17)	(13,17)	(14,17)	(15,17)	(16,17)	(17,17)	(18,17)	(19,17)	(20,17)
(1,18)	(2,18)	(3,18)	(4,18)	(5,18)	(6,18)	(7,18)	(8,18)	(9,18)	(10,18)	(11,18)	(12,18)	(13,18)	(14,18)	(15,18)	(16,18)	(17,18)	(18,18)	(19,18)	(20,18)
(1,19)	(2,19)	(3,19)	(4,19)	(5,19)	(6,19)	(7,19)	(8,19)	(9,19)	(10,19)	(11,19)	(12,19)	(13,19)	(14,19)	(15,19)	(16,19)	(17,19)	(18,19)	(19,19)	(20,19)
(1,20)	(2,20)	(3,20)	(4,20)	(5,20)	(6,20)	(7,20)	(8,20)	(9,20)	(10,20)	(11,20)	(12,20)	(13,20)	(14,20)	(15,20)	(16,20)	(17,20)	(18,20)	(19,20)	(20,20)

Pr{(1,1) shows} = Pr{1 shows from 1^st D20}*Pr{1 shows from 2^nd D20} = (1/20)*(1/20) = 1/400 @ 0.0025

In random samples of 100 tosses of the pair of dice, we expect approximately 100*Pr{(1,1)} = 100*(1/400) @ 0.25

The smallest sample size for which we expect to observe one or more tosses showing the pair (1,1) is 1/(1/400) = 400.

Pair of Fair Dice, one with face values {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24} and one with face values {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}

(1,1)	(2,1)	(3,1)	(4,1)	(5,1)	(6,1)	(7,1)	(8,1)	(9,1)	(10,1)	(11,1)	(12,1)	(13,1)	(14,1)	(15,1)	(16,1)	(17,1)	(18,1)	(19,1)	(21,1)	(22,1)	(23,1)	(24,1)
(1,2)	(2,2)	(3,2)	(4,2)	(5,2)	(6,2)	(7,2)	(8,2)	(9,2)	(10,2)	(11,2)	(12,2)	(13,2)	(14,2)	(15,2)	(16,2)	(17,2)	(18,2)	(19,2)	(21,2)	(22,2)	(23,2)	(24,2)
(1,3)	(2,3)	(3,3)	(4,3)	(5,3)	(6,3)	(7,3)	(8,3)	(9,3)	(10,3)	(11,3)	(12,3)	(13,3)	(14,3)	(15,3)	(16,3)	(17,3)	(18,3)	(19,3)	(21,3)	(22,3)	(23,3)	(24,3)
(1,4)	(2,4)	(3,4)	(4,4)	(5,4)	(6,4)	(7,4)	(8,4)	(9,4)	(10,4)	(11,4)	(12,4)	(13,4)	(14,4)	(15,4)	(16,4)	(17,4)	(18,4)	(19,4)	(21,4)	(22,4)	(23,4)	(24,4)
(1,5)	(2,5)	(3,5)	(4,5)	(5,5)	(6,5)	(7,5)	(8,5)	(9,5)	(10,5)	(11,5)	(12,5)	(13,5)	(14,5)	(15,5)	(16,5)	(17,5)	(18,5)	(19,5)	(21,5)	(22,5)	(23,5)	(24,5)
(1,6)	(2,6)	(3,6)	(4,6)	(5,6)	(6,6)	(7,6)	(8,6)	(9,6)	(10,6)	(11,6)	(12,6)	(13,6)	(14,6)	(15,6)	(16,6)	(17,6)	(18,6)	(19,6)	(21,6)	(22,6)	(23,6)	(24,6)
(1,7)	(2,7)	(3,7)	(4,7)	(5,7)	(6,7)	(71,7)	(8,7)	(9,7)	(10,7)	(11,7)	(12,7)	(13,7)	(14,7)	(15,7)	(16,7)	(17,7)	(18,7)	(19,7)	(21,7)	(22,7)	(23,7)	(24,7)
(1,8)	(2,8)	(3,8)	(4,8)	(5,8)	(6,8)	(7,8)	(8,8)	(9,8)	(10,8)	(11,8)	(12,8)	(13,8)	(14,8)	(15,8)	(16,8)	(17,8)	(18,8)	(19,8)	(21,8)	(22,8)	(23,8)	(24,8)
(1,9)	(2,9)	(3,9)	(4,9)	(5,9)	(6,9)	(7,9)	(8,9)	(9,9)	(10,9)	(11,9)	(12,9)	(13,9)	(14,9)	(15,9)	(16,9)	(17,9)	(18,9)	(19,9)	(21,9)	(22,9)	(23,9)	(24,9)
(1,10)	(2,10)	(3,10)	(4,10)	(5,10)	(6,10)	(7,10)	(8,10)	(9,10)	(10,10)	(11,10)	(12,10)	(13,10)	(14,10)	(15,10)	(16,10)	(17,10)	(18,10)	(19,10)	(21,10)	(22,10)	(23,10)	(24,10)
(1,11)	(2,11)	(3,11)	(4,11)	(5,11)	(6,11)	(7,11)	(8,11)	(9,11)	(10,11)	(11,11)	(12,11)	(13,11)	(14,11)	(15,11)	(16,11)	(17,11)	(18,11)	(19,11)	(21,11)	(22,11)	(23,11)	(24,11)
(1,12)	(2,12)	(3,12)	(4,12)	(5,12)	(6,12)	(7,12)	(8,12)	(9,12)	(10,12)	(11,12)	(12,12)	(13,12)	(14,12)	(15,12)	(16,12)	(17,12)	(18,12)	(19,12)	(21,12)	(22,12)	(23,12)	(24,12)
(1,13)	(2,13)	(3,13)	(4,13)	(5,13)	(6,13)	(7,13)	(8,13)	(9,13)	(10,13)	(11,13)	(12,13)	(13,13)	(14,13)	(15,13)	(16,13)	(17,13)	(18,13)	(19,13)	(21,13)	(22,13)	(23,13)	(24,13)
(1,14)	(2,14)	(3,14)	(4,14)	(5,14)	(6,14)	(7,14)	(8,14)	(9,14)	(10,14)	(11,14)	(12,14)	(13,14)	(14,14)	(15,14)	(16,14)	(17,14)	(18,14)	(19,14)	(21,14)	(22,14)	(23,14)	(24,14)
(1,15)	(2,15)	(3,15)	(4,15)	(5,15)	(6,15)	(7,15)	(8,15)	(9,15)	(10,15)	(11,15)	(12,15)	(13,15)	(14,15)	(15,15)	(16,15)	(17,15)	(18,15)	(19,15)	(21,15)	(22,15)	(23,15)	(24,15)
(1,16)	(2,16)	(3,16)	(4,16)	(5,16)	(6,16)	(7,16)	(8,16)	(9,16)	(10,16)	(11,16)	(12,16)	(13,16)	(14,16)	(15,16)	(16,16)	(17,16)	(18,16)	(19,16)	(21,16)	(22,16)	(23,16)	(24,16)
(1,17)	(2,17)	(3,17)	(4,17)	(5,17)	(6,17)	(71,17)	(8,17)	(9,17)	(10,17)	(11,17)	(12,17)	(13,17)	(14,17)	(15,17)	(16,17)	(17,17)	(18,17)	(19,17)	(21,17)	(22,17)	(23,17)	(24,17)
(1,18)	(2,18)	(3,18)	(4,18)	(5,18)	(6,18)	(7,18)	(8,18)	(9,18)	(10,18)	(11,18)	(12,18)	(13,18)	(14,18)	(15,18)	(16,18)	(17,18)	(18,18)	(19,18)	(21,18)	(22,18)	(23,18)	(24,18)
(1,19)	(2,19)	(3,19)	(4,19)	(5,19)	(6,19)	(7,19)	(8,19)	(9,19)	(10,19)	(11,19)	(12,19)	(13,19)	(14,19)	(15,19)	(16,19)	(17,19)	(18,19)	(19,19)	(21,19)	(22,19)	(23,19)	(24,19)
(1,20)	(2,20)	(3,20)	(4,20)	(5,20)	(6,20)	(7,20)	(8,20)	(9,20)	(10,20)	(11,20)	(12,20)	(13,20)	(14,20)	(15,20)	(16,20)	(17,20)	(18,20)	(19,20)	(21,20)	(22,20)	(23,20)	(24,20)
(21,1)	(22,1)	(23,1)	(24,1)
(21,2)	(22,2)	(23,2)	(24,2)
(21,3)	(22,3)	(23,3)	(24,3)
(21,4)	(22,4)	(23,4)	(24,4)
(21,5)	(22,5)	(23,5)	(24,5)
(21,6)	(22,6)	(23,6)	(24,6)
(21,7)	(22,7)	(23,7)	(24,7)
(21,8)	(22,8)	(23,8)	(24,8)
(21,9)	(22,9)	(23,9)	(24,9)
(21,10)	(22,10)	(23,10)	(24,10)
(21,11)	(22,11)	(23,11)	(24,11)
(21,12)	(22,12)	(23,12)	(24,12)
(21,13)	(22,13)	(23,13)	(24,13)
(21,14)	(22,14)	(23,14)	(24,14)
(21,15)	(22,15)	(23,15)	(24,15)
(21,16)	(22,16)	(23,16)	(24,16)
(21,17)	(22,17)	(23,17)	(24,17)
(21,18)	(22,18)	(23,18)	(24,18)
(21,19)	(22,19)	(23,19)	(24,19)
(21,20)	(22,20)	(23,20)	(24,20)

Pr{(1,1) shows} = Pr{1 shows from 1^st D24}*Pr{1 shows from 2^nd D20} = (1/24)*(1/20) = 1/480 @ 0.0021

In random samples of 100 tosses of the pair of dice, we expect approximately 100*Pr{(1,1)} = 100*(1/480) @ 0.21

The smallest sample size for which we expect to observe one or more tosses showing the pair (1,1) is 1/(1/480) = 480.

Pair of Fair Dice, each with face values {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24}

(1,1)	(2,1)	(3,1)	(4,1)	(5,1)	(6,1)	(7,1)	(8,1)	(9,1)	(10,1)	(11,1)	(12,1)	(13,1)	(14,1)	(15,1)	(16,1)	(17,1)	(18,1)	(19,1)	(21,1)	(22,1)	(23,1)	(24,1)
(1,2)	(2,2)	(3,2)	(4,2)	(5,2)	(6,2)	(7,2)	(8,2)	(9,2)	(10,2)	(11,2)	(12,2)	(13,2)	(14,2)	(15,2)	(16,2)	(17,2)	(18,2)	(19,2)	(21,2)	(22,2)	(23,2)	(24,2)
(1,3)	(2,3)	(3,3)	(4,3)	(5,3)	(6,3)	(7,3)	(8,3)	(9,3)	(10,3)	(11,3)	(12,3)	(13,3)	(14,3)	(15,3)	(16,3)	(17,3)	(18,3)	(19,3)	(21,3)	(22,3)	(23,3)	(24,3)
(1,4)	(2,4)	(3,4)	(4,4)	(5,4)	(6,4)	(7,4)	(8,4)	(9,4)	(10,4)	(11,4)	(12,4)	(13,4)	(14,4)	(15,4)	(16,4)	(17,4)	(18,4)	(19,4)	(21,4)	(22,4)	(23,4)	(24,4)
(1,5)	(2,5)	(3,5)	(4,5)	(5,5)	(6,5)	(7,5)	(8,5)	(9,5)	(10,5)	(11,5)	(12,5)	(13,5)	(14,5)	(15,5)	(16,5)	(17,5)	(18,5)	(19,5)	(21,5)	(22,5)	(23,5)	(24,5)
(1,6)	(2,6)	(3,6)	(4,6)	(5,6)	(6,6)	(7,6)	(8,6)	(9,6)	(10,6)	(11,6)	(12,6)	(13,6)	(14,6)	(15,6)	(16,6)	(17,6)	(18,6)	(19,6)	(21,6)	(22,6)	(23,6)	(24,6)
(1,7)	(2,7)	(3,7)	(4,7)	(5,7)	(6,7)	(71,7)	(8,7)	(9,7)	(10,7)	(11,7)	(12,7)	(13,7)	(14,7)	(15,7)	(16,7)	(17,7)	(18,7)	(19,7)	(21,7)	(22,7)	(23,7)	(24,7)
(1,8)	(2,8)	(3,8)	(4,8)	(5,8)	(6,8)	(7,8)	(8,8)	(9,8)	(10,8)	(11,8)	(12,8)	(13,8)	(14,8)	(15,8)	(16,8)	(17,8)	(18,8)	(19,8)	(21,8)	(22,8)	(23,8)	(24,8)
(1,9)	(2,9)	(3,9)	(4,9)	(5,9)	(6,9)	(7,9)	(8,9)	(9,9)	(10,9)	(11,9)	(12,9)	(13,9)	(14,9)	(15,9)	(16,9)	(17,9)	(18,9)	(19,9)	(21,9)	(22,9)	(23,9)	(24,9)
(1,10)	(2,10)	(3,10)	(4,10)	(5,10)	(6,10)	(7,10)	(8,10)	(9,10)	(10,10)	(11,10)	(12,10)	(13,10)	(14,10)	(15,10)	(16,10)	(17,10)	(18,10)	(19,10)	(21,10)	(22,10)	(23,10)	(24,10)
(1,11)	(2,11)	(3,11)	(4,11)	(5,11)	(6,11)	(7,11)	(8,11)	(9,11)	(10,11)	(11,11)	(12,11)	(13,11)	(14,11)	(15,11)	(16,11)	(17,11)	(18,11)	(19,11)	(21,11)	(22,11)	(23,11)	(24,11)
(1,12)	(2,12)	(3,12)	(4,12)	(5,12)	(6,12)	(7,12)	(8,12)	(9,12)	(10,12)	(11,12)	(12,12)	(13,12)	(14,12)	(15,12)	(16,12)	(17,12)	(18,12)	(19,12)	(21,12)	(22,12)	(23,12)	(24,12)
(1,13)	(2,13)	(3,13)	(4,13)	(5,13)	(6,13)	(7,13)	(8,13)	(9,13)	(10,13)	(11,13)	(12,13)	(13,13)	(14,13)	(15,13)	(16,13)	(17,13)	(18,13)	(19,13)	(21,13)	(22,13)	(23,13)	(24,13)
(1,14)	(2,14)	(3,14)	(4,14)	(5,14)	(6,14)	(7,14)	(8,14)	(9,14)	(10,14)	(11,14)	(12,14)	(13,14)	(14,14)	(15,14)	(16,14)	(17,14)	(18,14)	(19,14)	(21,14)	(22,14)	(23,14)	(24,14)
(1,15)	(2,15)	(3,15)	(4,15)	(5,15)	(6,15)	(7,15)	(8,15)	(9,15)	(10,15)	(11,15)	(12,15)	(13,15)	(14,15)	(15,15)	(16,15)	(17,15)	(18,15)	(19,15)	(21,15)	(22,15)	(23,15)	(24,15)
(1,16)	(2,16)	(3,16)	(4,16)	(5,16)	(6,16)	(7,16)	(8,16)	(9,16)	(10,16)	(11,16)	(12,16)	(13,16)	(14,16)	(15,16)	(16,16)	(17,16)	(18,16)	(19,16)	(21,16)	(22,16)	(23,16)	(24,16)
(1,17)	(2,17)	(3,17)	(4,17)	(5,17)	(6,17)	(71,17)	(8,17)	(9,17)	(10,17)	(11,17)	(12,17)	(13,17)	(14,17)	(15,17)	(16,17)	(17,17)	(18,17)	(19,17)	(21,17)	(22,17)	(23,17)	(24,17)
(1,18)	(2,18)	(3,18)	(4,18)	(5,18)	(6,18)	(7,18)	(8,18)	(9,18)	(10,18)	(11,18)	(12,18)	(13,18)	(14,18)	(15,18)	(16,18)	(17,18)	(18,18)	(19,18)	(21,18)	(22,18)	(23,18)	(24,18)
(1,19)	(2,19)	(3,19)	(4,19)	(5,19)	(6,19)	(7,19)	(8,19)	(9,19)	(10,19)	(11,19)	(12,19)	(13,19)	(14,19)	(15,19)	(16,19)	(17,19)	(18,19)	(19,19)	(21,19)	(22,19)	(23,19)	(24,19)
(1,20)	(2,20)	(3,20)	(4,20)	(5,20)	(6,20)	(7,20)	(8,20)	(9,20)	(10,20)	(11,20)	(12,20)	(13,20)	(14,20)	(15,20)	(16,20)	(17,20)	(18,20)	(19,20)	(21,20)	(22,20)	(23,20)	(24,20)
(1,21)	(2,21)	(3,21)	(4,21)	(5,21)	(6,21)	(7,21)	(8,21)	(9,21)	(10,21)	(11,21)	(12,21)	(13,21)	(14,21)	(15,21)	(16,21)	(17,21)	(18,21)	(19,21)	(21,21)	(22,21)	(23,21)	(24,21)
(1,22)	(2,22)	(3,22)	(4,22)	(5,22)	(6,22)	(7,22)	(8,22)	(9,22)	(10,22)	(11,22)	(12,22)	(13,22)	(14,22)	(15,22)	(16,22)	(17,22)	(18,22)	(19,22)	(21,22)	(22,22)	(23,22)	(24,22)
(1,23)	(2,23)	(3,23)	(4,23)	(5,23)	(6,23)	(7,23)	(8,23)	(9,23)	(10,23)	(11,23)	(12,23)	(13,23)	(14,23)	(15,23)	(16,23)	(17,23)	(18,23)	(19,23)	(21,23)	(22,23)	(23,23)	(24,23)
(1,24)	(2,24)	(3,24)	(4,24)	(5,24)	(6,24)	(7,24)	(8,24)	(9,24)	(10,24)	(11,24)	(12,24)	(13,24)	(14,24)	(15,24)	(16,24)	(17,24)	(18,24)	(19,24)	(21,24)	(22,24)	(23,24)	(24,24)

Pr{(1,1) shows} = Pr{1 shows from 1^st D24}*Pr{1 shows from 2^nd D24} = (1/24)*(1/24) = 1/576 @ 0.0017

In random samples of 100 tosses of the pair of dice, we expect approximately 100*Pr{(1,1)} = 100*(1/576) @ 0.17

The smallest sample size for which we expect to observe one or more tosses showing the pair (1,1) is 1/(1/576) = 576.

Pair of Fair Dice, one with face values {1,2,3,4,5,6,7,8,9,10,11,27,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30} and one with face values {1,2,3,4,5,6,7,8,9,10,11,27,13,14,15,16,17,18,19,24}

(1,1)	(2,1)	(3,1)	(4,1)	(5,1)	(6,1)	(7,1)	(8,1)	(9,1)	(10,1)	(11,1)	(12,1)	(13,1)	(14,1)	(15,1)	(16,1)	(17,1)	(18,1)	(19,1)	(20,1)
(1,2)	(2,2)	(3,2)	(4,2)	(5,2)	(6,2)	(7,2)	(8,2)	(9,2)	(10,2)	(11,2)	(12,2)	(13,2)	(14,2)	(15,2)	(16,2)	(17,2)	(18,2)	(19,2)	(20,2)
(1,3)	(2,3)	(3,3)	(4,3)	(5,3)	(6,3)	(7,3)	(8,3)	(9,3)	(10,3)	(11,3)	(12,3)	(13,3)	(14,3)	(15,3)	(16,3)	(17,3)	(18,3)	(19,3)	(20,3)
(1,4)	(2,4)	(3,4)	(4,4)	(5,4)	(6,4)	(7,4)	(8,4)	(9,4)	(10,4)	(11,4)	(12,4)	(13,4)	(14,4)	(15,4)	(16,4)	(17,4)	(18,4)	(19,4)	(20,4)
(1,5)	(2,5)	(3,5)	(4,5)	(5,5)	(6,5)	(7,5)	(8,5)	(9,5)	(10,5)	(11,5)	(12,5)	(13,5)	(14,5)	(15,5)	(16,5)	(17,5)	(18,5)	(19,5)	(20,5)
(1,6)	(2,6)	(3,6)	(4,6)	(5,6)	(6,6)	(7,6)	(8,6)	(9,6)	(10,6)	(11,6)	(12,6)	(13,6)	(14,6)	(15,6)	(16,6)	(17,6)	(18,6)	(19,6)	(20,6)
(1,7)	(2,7)	(3,7)	(4,7)	(5,7)	(6,7)	(71,7)	(8,7)	(9,7)	(10,7)	(11,7)	(12,7)	(13,7)	(14,7)	(15,7)	(16,7)	(17,7)	(18,7)	(19,7)	(20,7)
(1,8)	(2,8)	(3,8)	(4,8)	(5,8)	(6,8)	(7,8)	(8,8)	(9,8)	(10,8)	(11,8)	(12,8)	(13,8)	(14,8)	(15,8)	(16,8)	(17,8)	(18,8)	(19,8)	(20,8)
(1,9)	(2,9)	(3,9)	(4,9)	(5,9)	(6,9)	(7,9)	(8,9)	(9,9)	(10,9)	(11,9)	(12,9)	(13,9)	(14,9)	(15,9)	(16,9)	(17,9)	(18,9)	(19,9)	(20,9)
(1,10)	(2,10)	(3,10)	(4,10)	(5,10)	(6,10)	(7,10)	(8,10)	(9,10)	(10,10)	(11,10)	(12,10)	(13,10)	(14,10)	(15,10)	(16,10)	(17,10)	(18,10)	(19,10)	(20,10)
(1,11)	(2,11)	(3,11)	(4,11)	(5,11)	(6,11)	(7,11)	(8,11)	(9,11)	(10,11)	(11,11)	(12,11)	(13,11)	(14,11)	(15,11)	(16,11)	(17,11)	(18,11)	(19,11)	(20,11)
(1,12)	(2,12)	(3,12)	(4,12)	(5,12)	(6,12)	(7,12)	(8,12)	(9,12)	(10,12)	(11,12)	(12,12)	(13,12)	(14,12)	(15,12)	(16,12)	(17,12)	(18,12)	(19,12)	(20,12)
(1,13)	(2,13)	(3,13)	(4,13)	(5,13)	(6,13)	(7,13)	(8,13)	(9,13)	(10,13)	(11,13)	(12,13)	(13,13)	(14,13)	(15,13)	(16,13)	(17,13)	(18,13)	(19,13)	(20,13)
(1,14)	(2,14)	(3,14)	(4,14)	(5,14)	(6,14)	(7,14)	(8,14)	(9,14)	(10,14)	(11,14)	(12,14)	(13,14)	(14,14)	(15,14)	(16,14)	(17,14)	(18,14)	(19,14)	(20,14)
(1,15)	(2,15)	(3,15)	(4,15)	(5,15)	(6,15)	(7,15)	(8,15)	(9,15)	(10,15)	(11,15)	(12,15)	(13,15)	(14,15)	(15,15)	(16,15)	(17,15)	(18,15)	(19,15)	(20,15)
(1,16)	(2,16)	(3,16)	(4,16)	(5,16)	(6,16)	(7,16)	(8,16)	(9,16)	(10,16)	(11,16)	(12,16)	(13,16)	(14,16)	(15,16)	(16,16)	(17,16)	(18,16)	(19,16)	(20,16)
(1,17)	(2,17)	(3,17)	(4,17)	(5,17)	(6,17)	(71,17)	(8,17)	(9,17)	(10,17)	(11,17)	(12,17)	(13,17)	(14,17)	(15,17)	(16,17)	(17,17)	(18,17)	(19,17)	(20,17)
(1,18)	(2,18)	(3,18)	(4,18)	(5,18)	(6,18)	(7,18)	(8,18)	(9,18)	(10,18)	(11,18)	(12,18)	(13,18)	(14,18)	(15,18)	(16,18)	(17,18)	(18,18)	(19,18)	(20,18)
(1,19)	(2,19)	(3,19)	(4,19)	(5,19)	(6,19)	(7,19)	(8,19)	(9,19)	(10,19)	(11,19)	(12,19)	(13,19)	(14,19)	(15,19)	(16,19)	(17,19)	(18,19)	(19,19)	(20,19)
(1,20)	(2,20)	(3,20)	(4,20)	(5,20)	(6,20)	(7,20)	(8,20)	(9,20)	(10,20)	(11,20)	(12,20)	(13,20)	(14,20)	(15,20)	(16,20)	(17,20)	(18,20)	(19,20)	(20,20)
(1,21)	(2,21)	(3,21)	(4,21)	(5,21)	(6,21)	(7,21)	(8,21)	(9,21)	(10,21)	(11,21)	(12,21)	(13,21)	(14,21)	(15,21)	(16,21)	(17,21)	(18,21)	(19,21)	(20,21)
(1,22)	(2,22)	(3,22)	(4,22)	(5,22)	(6,22)	(7,22)	(8,22)	(9,22)	(10,22)	(11,22)	(12,22)	(13,22)	(14,22)	(15,22)	(16,22)	(17,22)	(18,22)	(19,22)	(20,22)
(1,23)	(2,23)	(3,23)	(4,23)	(5,23)	(6,23)	(7,23)	(8,23)	(9,23)	(10,23)	(11,23)	(12,23)	(13,23)	(14,23)	(15,23)	(16,23)	(17,23)	(18,23)	(19,23)	(20,23)
(1,24)	(2,24)	(3,24)	(4,24)	(5,24)	(6,24)	(7,24)	(8,24)	(9,24)	(10,24)	(11,24)	(12,24)	(13,24)	(14,24)	(15,24)	(16,24)	(17,24)	(18,24)	(19,24)	(20,24)
(21,1)	(22,1)	(23,1)	(24,1)	(25,1)	(26,1)	(27,1)	(28,1)	(29,1)	(30,1)
(21,2)	(22,2)	(23,2)	(24,2)	(25,2)	(26,2)	(27,2)	(28,2)	(29,2)	(30,2)
(21,3)	(22,3)	(23,3)	(24,3)	(25,3)	(26,3)	(27,3)	(28,3)	(29,3)	(30,3)
(21,4)	(22,4)	(23,4)	(24,4)	(25,4)	(26,4)	(27,4)	(28,4)	(29,4)	(30,4)
(21,5)	(22,5)	(23,5)	(24,5)	(25,5)	(26,5)	(27,5)	(28,5)	(29,5)	(30,5)
(21,6)	(22,6)	(23,6)	(24,6)	(25,6)	(26,6)	(27,6)	(28,6)	(29,6)	(30,6)
(21,7)	(22,7)	(23,7)	(24,7)	(25,7)	(26,7)	(27,7)	(28,7)	(29,7)	(30,7)
(21,8)	(22,8)	(23,8)	(24,8)	(25,8)	(26,8)	(27,8)	(28,8)	(29,8)	(30,8)
(21,9)	(22,9)	(23,9)	(24,9)	(25,9)	(26,9)	(27,9)	(28,9)	(29,9)	(30,9)
(21,10)	(22,10)	(23,10)	(24,10)	(25,10)	(26,10)	(27,10)	(28,10)	(29,10)	(30,10)
(21,11)	(22,11)	(23,11)	(24,11)	(25,11)	(26,11)	(27,11)	(28,11)	(29,11)	(30,11)
(21,12)	(22,12)	(23,12)	(24,12)	(25,12)	(26,12)	(27,12)	(28,12)	(29,12)	(30,12)
(21,13)	(22,13)	(23,13)	(24,13)	(25,13)	(26,13)	(27,13)	(28,13)	(29,13)	(30,13)
(21,14)	(22,14)	(23,14)	(24,14)	(25,14)	(26,14)	(27,14)	(28,14)	(29,14)	(30,14)
(21,15)	(22,15)	(23,15)	(24,15)	(25,15)	(26,15)	(27,15)	(28,15)	(29,15)	(30,15)
(21,16)	(22,16)	(23,16)	(24,16)	(25,16)	(26,16)	(27,16)	(28,16)	(29,16)	(30,16)
(21,17)	(22,17)	(23,17)	(24,17)	(25,17)	(26,17)	(27,17)	(28,17)	(29,17)	(30,17)
(21,18)	(22,18)	(23,18)	(24,18)	(25,18)	(26,18)	(27,18)	(28,18)	(29,18)	(30,18)
(21,19)	(22,19)	(23,19)	(24,19)	(25,19)	(26,19)	(27,19)	(28,19)	(29,19)	(30,19)
(21,20)	(22,20)	(23,20)	(24,20)	(25,20)	(26,20)	(27,20)	(28,20)	(29,20)	(30,20)
(21,21)	(22,21)	(23,21)	(24,21)	(25,21)	(26,21)	(27,21)	(28,21)	(29,21)	(30,21)
(21,22)	(22,22)	(23,22)	(24,22)	(25,22)	(26,22)	(27,22)	(28,22)	(29,22)	(30,22)
(21,23)	(22,23)	(23,23)	(24,23)	(25,23)	(26,23)	(27,23)	(28,23)	(29,23)	(30,23)
(21,24)	(22,24)	(23,24)	(24,24)	(25,24)	(26,24)	(27,24)	(28,24)	(29,24)	(30,24)

Pr{(1,1) shows} = Pr{1 shows from 1^st D30}*Pr{1 shows from 2^nd D24} = (1/30)*(1/24) = 1/720 @ 0.0014

In random samples of 100 tosses of the pair of dice, we expect approximately 100*Pr{(1,1)} = 100*(1/720) @ 0.14The smallest sample size for which we expect to observe one or more tosses showing the pair (1,1) is 1/(1/720) = 720.

Pair of Fair Dice, each with face values {1,2,3,4,5,6,7,8,9,10,11,27,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30}

(1,1)	(2,1)	(3,1)	(4,1)	(5,1)	(6,1)	(7,1)	(8,1)	(9,1)	(10,1)	(11,1)	(12,1)	(13,1)	(14,1)	(15,1)	(16,1)	(17,1)	(18,1)	(19,1)	(20,1)
(1,2)	(2,2)	(3,2)	(4,2)	(5,2)	(6,2)	(7,2)	(8,2)	(9,2)	(10,2)	(11,2)	(12,2)	(13,2)	(14,2)	(15,2)	(16,2)	(17,2)	(18,2)	(19,2)	(20,2)
(1,3)	(2,3)	(3,3)	(4,3)	(5,3)	(6,3)	(7,3)	(8,3)	(9,3)	(10,3)	(11,3)	(12,3)	(13,3)	(14,3)	(15,3)	(16,3)	(17,3)	(18,3)	(19,3)	(20,3)
(1,4)	(2,4)	(3,4)	(4,4)	(5,4)	(6,4)	(7,4)	(8,4)	(9,4)	(10,4)	(11,4)	(12,4)	(13,4)	(14,4)	(15,4)	(16,4)	(17,4)	(18,4)	(19,4)	(20,4)
(1,5)	(2,5)	(3,5)	(4,5)	(5,5)	(6,5)	(7,5)	(8,5)	(9,5)	(10,5)	(11,5)	(12,5)	(13,5)	(14,5)	(15,5)	(16,5)	(17,5)	(18,5)	(19,5)	(20,5)
(1,6)	(2,6)	(3,6)	(4,6)	(5,6)	(6,6)	(7,6)	(8,6)	(9,6)	(10,6)	(11,6)	(12,6)	(13,6)	(14,6)	(15,6)	(16,6)	(17,6)	(18,6)	(19,6)	(20,6)
(1,7)	(2,7)	(3,7)	(4,7)	(5,7)	(6,7)	(71,7)	(8,7)	(9,7)	(10,7)	(11,7)	(12,7)	(13,7)	(14,7)	(15,7)	(16,7)	(17,7)	(18,7)	(19,7)	(20,7)
(1,8)	(2,8)	(3,8)	(4,8)	(5,8)	(6,8)	(7,8)	(8,8)	(9,8)	(10,8)	(11,8)	(12,8)	(13,8)	(14,8)	(15,8)	(16,8)	(17,8)	(18,8)	(19,8)	(20,8)
(1,9)	(2,9)	(3,9)	(4,9)	(5,9)	(6,9)	(7,9)	(8,9)	(9,9)	(10,9)	(11,9)	(12,9)	(13,9)	(14,9)	(15,9)	(16,9)	(17,9)	(18,9)	(19,9)	(20,9)
(1,10)	(2,10)	(3,10)	(4,10)	(5,10)	(6,10)	(7,10)	(8,10)	(9,10)	(10,10)	(11,10)	(12,10)	(13,10)	(14,10)	(15,10)	(16,10)	(17,10)	(18,10)	(19,10)	(20,10)
(1,11)	(2,11)	(3,11)	(4,11)	(5,11)	(6,11)	(7,11)	(8,11)	(9,11)	(10,11)	(11,11)	(12,11)	(13,11)	(14,11)	(15,11)	(16,11)	(17,11)	(18,11)	(19,11)	(20,11)
(1,12)	(2,12)	(3,12)	(4,12)	(5,12)	(6,12)	(7,12)	(8,12)	(9,12)	(10,12)	(11,12)	(12,12)	(13,12)	(14,12)	(15,12)	(16,12)	(17,12)	(18,12)	(19,12)	(20,12)
(1,13)	(2,13)	(3,13)	(4,13)	(5,13)	(6,13)	(7,13)	(8,13)	(9,13)	(10,13)	(11,13)	(12,13)	(13,13)	(14,13)	(15,13)	(16,13)	(17,13)	(18,13)	(19,13)	(20,13)
(1,14)	(2,14)	(3,14)	(4,14)	(5,14)	(6,14)	(7,14)	(8,14)	(9,14)	(10,14)	(11,14)	(12,14)	(13,14)	(14,14)	(15,14)	(16,14)	(17,14)	(18,14)	(19,14)	(20,14)
(1,15)	(2,15)	(3,15)	(4,15)	(5,15)	(6,15)	(7,15)	(8,15)	(9,15)	(10,15)	(11,15)	(12,15)	(13,15)	(14,15)	(15,15)	(16,15)	(17,15)	(18,15)	(19,15)	(20,15)
(1,16)	(2,16)	(3,16)	(4,16)	(5,16)	(6,16)	(7,16)	(8,16)	(9,16)	(10,16)	(11,16)	(12,16)	(13,16)	(14,16)	(15,16)	(16,16)	(17,16)	(18,16)	(19,16)	(20,16)
(1,17)	(2,17)	(3,17)	(4,17)	(5,17)	(6,17)	(71,17)	(8,17)	(9,17)	(10,17)	(11,17)	(12,17)	(13,17)	(14,17)	(15,17)	(16,17)	(17,17)	(18,17)	(19,17)	(20,17)
(1,18)	(2,18)	(3,18)	(4,18)	(5,18)	(6,18)	(7,18)	(8,18)	(9,18)	(10,18)	(11,18)	(12,18)	(13,18)	(14,18)	(15,18)	(16,18)	(17,18)	(18,18)	(19,18)	(20,18)
(1,19)	(2,19)	(3,19)	(4,19)	(5,19)	(6,19)	(7,19)	(8,19)	(9,19)	(10,19)	(11,19)	(12,19)	(13,19)	(14,19)	(15,19)	(16,19)	(17,19)	(18,19)	(19,19)	(20,19)
(1,20)	(2,20)	(3,20)	(4,20)	(5,20)	(6,20)	(7,20)	(8,20)	(9,20)	(10,20)	(11,20)	(12,20)	(13,20)	(14,20)	(15,20)	(16,20)	(17,20)	(18,20)	(19,20)	(20,20)
(1,21)	(2,21)	(3,21)	(4,21)	(5,21)	(6,21)	(7,21)	(8,21)	(9,21)	(10,21)	(11,21)	(12,21)	(13,21)	(14,21)	(15,21)	(16,21)	(17,21)	(18,21)	(19,21)	(20,21)
(1,22)	(2,22)	(3,22)	(4,22)	(5,22)	(6,22)	(7,22)	(8,22)	(9,22)	(10,22)	(11,22)	(12,22)	(13,22)	(14,22)	(15,22)	(16,22)	(17,22)	(18,22)	(19,22)	(20,22)
(1,23)	(2,23)	(3,23)	(4,23)	(5,23)	(6,23)	(7,23)	(8,23)	(9,23)	(10,23)	(11,23)	(12,23)	(13,23)	(14,23)	(15,23)	(16,23)	(17,23)	(18,23)	(19,23)	(20,23)
(1,24)	(2,24)	(3,24)	(4,24)	(5,24)	(6,24)	(7,24)	(8,24)	(9,24)	(10,24)	(11,24)	(12,24)	(13,24)	(14,24)	(15,24)	(16,24)	(17,24)	(18,24)	(19,24)	(20,24)
(1,25)	(2,25)	(3,25)	(4,25)	(5,25)	(6,25)	(7,25)	(8,25)	(9,25)	(10,25)	(11,25)	(12,25)	(13,25)	(14,25)	(15,25)	(16,25)	(17,25)	(18,25)	(19,25)	(20,25)
(1,26)	(2,26)	(3,26)	(4,26)	(5,26)	(6,26)	(7,26)	(8,26)	(9,26)	(10,26)	(11,26)	(12,26)	(13,26)	(14,26)	(15,26)	(16,26)	(17,26)	(18,26)	(19,26)	(20,26)
(1,27)	(2,27)	(3,27)	(4,27)	(5,27)	(6,27)	(7,27)	(8,27)	(9,27)	(10,27)	(11,27)	(12,27)	(13,27)	(14,27)	(15,27)	(16,27)	(17,27)	(18,27)	(19,27)	(20,27)
(1,28)	(2,28)	(3,28)	(4,28)	(5,28)	(6,28)	(7,28)	(8,28)	(9,28)	(10,28)	(11,28)	(12,28)	(13,28)	(14,28)	(15,28)	(16,28)	(17,28)	(18,28)	(19,28)	(20,28)
(1,29)	(2,29)	(3,29)	(4,29)	(5,29)	(6,29)	(7,29)	(8,29)	(9,29)	(10,29)	(11,29)	(12,29)	(13,29)	(14,29)	(15,29)	(16,29)	(17,29)	(18,29)	(19,29)	(20,29)
(1,30)	(2,30)	(3,30)	(4,30)	(5,30)	(6,30)	(7,30)	(8,30)	(9,30)	(10,30)	(11,30)	(12,30)	(13,30)	(14,30)	(15,30)	(16,30)	(17,30)	(18,30)	(19,30)	(20,30)
(21,1)	(22,1)	(23,1)	(24,1)	(25,1)	(26,1)	(27,1)	(28,1)	(29,1)	(30,1)
(21,2)	(22,2)	(23,2)	(24,2)	(25,2)	(26,2)	(27,2)	(28,2)	(29,2)	(30,2)
(21,3)	(22,3)	(23,3)	(24,3)	(25,3)	(26,3)	(27,3)	(28,3)	(29,3)	(30,3)
(21,4)	(22,4)	(23,4)	(24,4)	(25,4)	(26,4)	(27,4)	(28,4)	(29,4)	(30,4)
(21,5)	(22,5)	(23,5)	(24,5)	(25,5)	(26,5)	(27,5)	(28,5)	(29,5)	(30,5)
(21,6)	(22,6)	(23,6)	(24,6)	(25,6)	(26,6)	(27,6)	(28,6)	(29,6)	(30,6)
(21,7)	(22,7)	(23,7)	(24,7)	(25,7)	(26,7)	(27,7)	(28,7)	(29,7)	(30,7)
(21,8)	(22,8)	(23,8)	(24,8)	(25,8)	(26,8)	(27,8)	(28,8)	(29,8)	(30,8)
(21,9)	(22,9)	(23,9)	(24,9)	(25,9)	(26,9)	(27,9)	(28,9)	(29,9)	(30,9)
(21,10)	(22,10)	(23,10)	(24,10)	(25,10)	(26,10)	(27,10)	(28,10)	(29,10)	(30,10)
(21,11)	(22,11)	(23,11)	(24,11)	(25,11)	(26,11)	(27,11)	(28,11)	(29,11)	(30,11)
(21,12)	(22,12)	(23,12)	(24,12)	(25,12)	(26,12)	(27,12)	(28,12)	(29,12)	(30,12)
(21,13)	(22,13)	(23,13)	(24,13)	(25,13)	(26,13)	(27,13)	(28,13)	(29,13)	(30,13)
(21,14)	(22,14)	(23,14)	(24,14)	(25,14)	(26,14)	(27,14)	(28,14)	(29,14)	(30,14)
(21,15)	(22,15)	(23,15)	(24,15)	(25,15)	(26,15)	(27,15)	(28,15)	(29,15)	(30,15)
(21,16)	(22,16)	(23,16)	(24,16)	(25,16)	(26,16)	(27,16)	(28,16)	(29,16)	(30,16)
(21,17)	(22,17)	(23,17)	(24,17)	(25,17)	(26,17)	(27,17)	(28,17)	(29,17)	(30,17)
(21,18)	(22,18)	(23,18)	(24,18)	(25,18)	(26,18)	(27,18)	(28,18)	(29,18)	(30,18)
(21,19)	(22,19)	(23,19)	(24,19)	(25,19)	(26,19)	(27,19)	(28,19)	(29,19)	(30,19)
(21,20)	(22,20)	(23,20)	(24,20)	(25,20)	(26,20)	(27,20)	(28,20)	(29,20)	(30,20)
(21,21)	(22,21)	(23,21)	(24,21)	(25,21)	(26,21)	(27,21)	(28,21)	(29,21)	(30,21)
(21,22)	(22,22)	(23,22)	(24,22)	(25,22)	(26,22)	(27,22)	(28,22)	(29,22)	(30,22)
(21,23)	(22,23)	(23,23)	(24,23)	(25,23)	(26,23)	(27,23)	(28,23)	(29,23)	(30,23)
(21,24)	(22,24)	(23,24)	(24,24)	(25,24)	(26,24)	(27,24)	(28,24)	(29,24)	(30,24)
(21,25)	(22,25)	(23,25)	(24,25)	(25,25)	(26,25)	(27,25)	(28,25)	(29,25)	(30,25)
(21,26)	(22,26)	(23,26)	(24,26)	(25,26)	(26,26)	(27,26)	(28,26)	(29,26)	(30,26)
(21,27)	(22,27)	(23,27)	(24,27)	(25,27)	(26,27)	(27,27)	(28,27)	(29,27)	(30,27)
(21,28)	(22,28)	(23,28)	(24,28)	(25,28)	(26,28)	(27,28)	(28,28)	(29,28)	(30,28)
(21,29)	(22,29)	(23,29)	(24,29)	(25,29)	(26,29)	(27,29)	(28,29)	(29,29)	(30,29)
(21,30)	(22,30)	(23,30)	(24,30)	(25,30)	(26,30)	(27,30)	(28,30)	(29,30)	(30,30)

Pr{(1,1) shows} = Pr{1 shows from 1^st D30}*Pr{1 shows from 2^nd D30} = (1/30)*(1/30) = 1/900 @ 0.001111111

In random samples of 100 tosses of the pair of dice, we expect approximately 100*Pr{(1,1)} = 100*(1/900) @ 0. 1111111

The smallest sample size for which we expect to observe one or more tosses showing the pair (1,1) is 1/(1/900) = 900.

Tracking The Pair (1,1) in Random Samples of n=100

Samples

1st Die	2nd Die	Pr{(1,1)}	Minimum n for E100=1 for (1,1)	E100	n	p
4	4	0.0625	16	6.25	7	0.07
5	5	0.04	25	4	11	0.11
8	8	0.015625	64	1.5625	0	0
10	10	0.01	100	1	0	0
10	12	0.008333333	120	0.8333	1	0
20	24	0.002083333	480	0.2083	0	0
30	30	0.001111111	900	0.1111	0	0

When an event is rare relative to a sample size, the occurrence of that event in samples of that size will be irregular.

At this point, work through all Part One (Fall and Spring) Case Types except Conditional Probability, unless you’re working ahead. Work one case type at a time. The only new cases left involve conditional probability.

A Partial List of Part One Case Types

Long Run Argument/Perfect Samples

Probability Rules

Color Slot Machine

Pairs of Dice

Random Variables

Next Case Types:

Conditional Probability

Clinical Trial Sketch

Design Fault Spot