7th
February 2010
Session
1.5
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
Pr
Pr
Pr
Pr
Pr
Pr
Pr
Pr
Pr
Pr
Pr
Pr
|
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
Pr
1/12
Pr
Pr
Pr
(1/12)+(1/12)=
2/12
Pr
Pr
Pr
(1/12)+(1/12)
+(1/12)=
3/12
Pr
Pr
Pr
(1/12)+(1/12)
+(1/12)=
3/12
Pr
Pr
Pr
(1/12)+(1/12)=
2/12
Pr
Pr
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
Pr
Pr
Pr
Pr
Pr
Pr
Pr
Pr
Pr
Pr
Pr
Pr
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
Pr
Pr
Pr
(3/60)+(8/60)=
11/60
Pr
Pr
Pr
(2/60)+(6/60)+(12/60)=
20/60
Pr
Pr
Pr
(1/60)+(4/60) +(9/60)=
14/60
Pr
Pr
Pr
8/60
Pr
Pr
3/60
Pr{Sum is a multiple of
3}=Pr{Sum is 3 or 6} = Pr{Sum=3} + Pr{sum=6} = (11/60) + (8/60) =19/60
Pr{Sum is 3 or 4 or 7} =
Pr{Sum is 3 } + Pr{Sum is 4} + Pr{Sum is 7} = (11/60) + (20/60) + (3/60) =
34/60
Compare sample
proportions (p) to model probabilities (P). Compare precision with increasing
sample size.
Samples:
6:30
|
Fair Pair Model |
|
|
|
Loaded Pair Model |
|
|
|
|
|
Sample #1 |
|
|
|
|
Sample #4 |
|
|
|
|
Sum |
n |
p |
P |
E200 |
n |
p |
P |
E200 |
|
2 |
22 |
0.110 |
0.083 |
16.7 |
14 |
0.070 |
0.067 |
13.3 |
|
3 |
36 |
0.180 |
0.167 |
33.3 |
44 |
0.220 |
0.183 |
36.7 |
|
4 |
49 |
0.245 |
0.250 |
50.0 |
70 |
0.350 |
0.333 |
66.7 |
|
5 |
50 |
0.250 |
0.250 |
50.0 |
40 |
0.200 |
0.233 |
46.7 |
|
6 |
24 |
0.120 |
0.167 |
33.3 |
23 |
0.115 |
0.133 |
26.7 |
|
7 |
19 |
0.095 |
0.083 |
16.7 |
9 |
0.045 |
0.050 |
10.0 |
|
Total |
200 |
1 |
1 |
200 |
200 |
1 |
1 |
200 |
|
Sample #2 |
|
|
|
|
Sample #5 |
|
|
|
|
Sum |
n |
p |
P |
E200 |
n |
p |
P |
E200 |
|
2 |
19 |
0.095 |
0.083 |
16.7 |
11 |
0.055 |
0.067 |
13.3 |
|
3 |
44 |
0.220 |
0.167 |
33.3 |
38 |
0.190 |
0.183 |
36.7 |
|
4 |
55 |
0.275 |
0.250 |
50.0 |
68 |
0.340 |
0.333 |
66.7 |
|
5 |
37 |
0.185 |
0.250 |
50.0 |
48 |
0.240 |
0.233 |
46.7 |
|
6 |
36 |
0.180 |
0.167 |
33.3 |
27 |
0.135 |
0.133 |
26.7 |
|
7 |
9 |
0.045 |
0.083 |
16.7 |
8 |
0.040 |
0.050 |
10.0 |
|
Total |
200 |
1 |
1 |
200 |
200 |
1 |
1 |
200 |
|
Sample #3 |
|
|
|
|
Sample #6 |
|
|
|
|
Sum |
n |
p |
P |
E200 |
n |
p |
P |
E200 |
|
2 |
15 |
0.075 |
0.083 |
16.7 |
16 |
0.080 |
0.067 |
13.3 |
|
3 |
43 |
0.215 |
0.167 |
33.3 |
36 |
0.180 |
0.183 |
36.7 |
|
4 |
50 |
0.250 |
0.250 |
50.0 |
60 |
0.300 |
0.333 |
66.7 |
|
5 |
58 |
0.290 |
0.250 |
50.0 |
51 |
0.255 |
0.233 |
46.7 |
|
6 |
26 |
0.130 |
0.167 |
33.3 |
28 |
0.140 |
0.133 |
26.7 |
|
7 |
8 |
0.040 |
0.083 |
16.7 |
9 |
0.045 |
0.050 |
10.0 |
|
Total |
200 |
1 |
1 |
200 |
200 |
1 |
1 |
200 |
|
Pooled 123 |
|
|
|
|
Pooled 456 |
|
|
|
|
Sum |
n |
p |
P |
E600 |
n |
p |
P |
E600 |
|
2 |
56 |
0.093 |
0.083 |
50 |
41 |
0.068 |
0.067 |
40 |
|
3 |
123 |
0.205 |
0.167 |
100 |
118 |
0.197 |
0.183 |
110 |
|
4 |
154 |
0.257 |
0.250 |
150 |
198 |
0.330 |
0.333 |
200 |
|
5 |
145 |
0.242 |
0.250 |
150 |
139 |
0.232 |
0.233 |
140 |
|
6 |
86 |
0.143 |
0.167 |
100 |
78 |
0.130 |
0.133 |
80 |
|
7 |
36 |
0.060 |
0.083 |
50 |
26 |
0.043 |
0.050 |
30 |
|
Total |
600 |
1 |
1 |
600 |
600 |
1 |
1 |
600 |
8:00
|
Fair Pair Model |
|
|
|
Loaded Pair Model |
|
|
|
|
|
Sample #1 |
|
|
|
|
Sample #4 |
|
|
|
|
Sum |
n |
p |
P |
E200 |
n |
p |
P |
E200 |
|
2 |
14 |
0.070 |
0.083 |
16.7 |
16 |
0.080 |
0.067 |
13.3 |
|
3 |
28 |
0.140 |
0.167 |
33.3 |
33 |
0.165 |
0.183 |
36.7 |
|
4 |
56 |
0.280 |
0.250 |
50.0 |
58 |
0.290 |
0.333 |
66.7 |
|
5 |
55 |
0.275 |
0.250 |
50.0 |
56 |
0.280 |
0.233 |
46.7 |
|
6 |
38 |
0.190 |
0.167 |
33.3 |
27 |
0.135 |
0.133 |
26.7 |
|
7 |
9 |
0.045 |
0.083 |
16.7 |
10 |
0.050 |
0.050 |
10.0 |
|
Total |
200 |
1 |
1 |
200 |
200 |
1 |
1 |
200 |
|
Sample #2 |
|
|
|
|
Sample #5 |
|
|
|
|
Sum |
n |
p |
P |
E200 |
n |
p |
P |
E200 |
|
2 |
12 |
0.060 |
0.083 |
16.7 |
12 |
0.060 |
0.067 |
13.3 |
|
3 |
38 |
0.190 |
0.167 |
33.3 |
33 |
0.165 |
0.183 |
36.7 |
|
4 |
44 |
0.220 |
0.250 |
50.0 |
80 |
0.400 |
0.333 |
66.7 |
|
5 |
63 |
0.315 |
0.250 |
50.0 |
45 |
0.225 |
0.233 |
46.7 |
|
6 |
28 |
0.140 |
0.167 |
33.3 |
23 |
0.115 |
0.133 |
26.7 |
|
7 |
15 |
0.075 |
0.083 |
16.7 |
7 |
0.035 |
0.050 |
10.0 |
|
Total |
200 |
1 |
1 |
200 |
200 |
1 |
1 |
200 |
|
Sample #3 |
|
|
|
|
Sample #6 |
|
|
|
|
Sum |
n |
p |
P |
E200 |
n |
p |
P |
E200 |
|
2 |
17 |
0.085 |
0.083 |
16.7 |
18 |
0.090 |
0.067 |
13.3 |
|
3 |
29 |
0.145 |
0.167 |
33.3 |
33 |
0.165 |
0.183 |
36.7 |
|
4 |
54 |
0.270 |
0.250 |
50.0 |
74 |
0.370 |
0.333 |
66.7 |
|
5 |
53 |
0.265 |
0.250 |
50.0 |
43 |
0.215 |
0.233 |
46.7 |
|
6 |
30 |
0.150 |
0.167 |
33.3 |
26 |
0.130 |
0.133 |
26.7 |
|
7 |
17 |
0.085 |
0.083 |
16.7 |
6 |
0.030 |
0.050 |
10.0 |
|
Total |
200 |
1 |
1 |
200 |
200 |
1 |
1 |
200 |
|
Pooled 123 |
|
|
|
|
Pooled 456 |
|
|
|
|
Sum |
n |
p |
P |
E600 |
n |
p |
P |
E600 |
|
2 |
43 |
0.072 |
0.083 |
50 |
46 |
0.077 |
0.067 |
40 |
|
3 |
95 |
0.158 |
0.167 |
100 |
99 |
0.165 |
0.183 |
110 |
|
4 |
154 |
0.257 |
0.250 |
150 |
212 |
0.353 |
0.333 |
200 |
|
5 |
171 |
0.285 |
0.250 |
150 |
144 |
0.240 |
0.233 |
140 |
|
6 |
96 |
0.160 |
0.167 |
100 |
76 |
0.127 |
0.133 |
80 |
|
7 |
41 |
0.068 |
0.083 |
50 |
23 |
0.038 |
0.050 |
30 |
|
Total |
600 |
1 |
1 |
600 |
600 |
1 |
1 |
600 |
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.
Case
Study #1.9
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 1,2,3,4) 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 |
There are 4*4=126distinct 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), (4,1), (4,2),
(4,3), (4,4)
|
(d4,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) |
Under the independent
multiplication principle,
Pr{(event from d4, event from d4)} = Pr
Pr
Pr
Pr
Pr
Pr
Pr
Pr
Pr
Pr
Pr
Pr
Pr
Pr{(1,4)} = Pr
Pr{(2,4)} = Pr
Pr{(3,4)} = Pr
Pr{(4,4)} = Pr
|
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 |
|
4 |
(1,4) → 5 |
(2,4) → 6 |
(3,4) → 7 |
(4,4) → 8 |
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
Pr
1/16
Pr
Pr
Pr
(1/16)+(1/16)=
2/16
Pr
Pr
Pr
(1/16)+(1/16)
+(1/16)=
3/16
Pr
Pr{One of (4,1),(2,3),(3,2),(1,4) Shows} =
Pr
(1/16)+(1/16)
+(1/16) + (1/16) =
4/16
Pr
Pr{One of (4,2),(3,3),(2,4) Shows} =
Pr
(1/16)+(1/16)+(1/16)=
3/16
Pr
Pr{(4,3),(3,4)} =
Pr{(4,3)}+Pr{(3,4)} =
(1/16)+ (1/16)=
2/16
Pr{Sum=8} =
Pr{(4,4)} =
1/16
Pr{Exactly One Odd Face} = Pr{ One of (1,2), (1,4), (3,2),
(3,4), (2,1), (4,1), (2,3), (4,3) Shows} =
Pr{(1,2)}+Pr{(1,4) }+Pr{ (3,2) }+Pr{ (3,4) }+Pr{ (2,1) }+Pr{
(4,1) }+Pr{ (2,3) }+Pr{ (4,3) } =
(1/16)+ (1/16)+ (1/16)+ (1/16)+ (1/16)+
(1/16)+ (1/16)+ (1/16) = 8/16
Pr{Sum is 3 or 8} =
Pr{Sum is 3} + Pr{Sum is 8} = Pr{One of (1,2) or (2,1) Shows} + Pr{(4,4)} =
Pr{(1,2)}+Pr{(2,1)}+Pr{(4,4)} = (1/16)+ (1/16)+ (1/16) = 3/16
Pr{Sum < 3 or Sum > 6} = Pr{Sum is 2 or 7 or 8} = Pr{One
of (1,2),(2,1),(3,4),(4,3),(4,4) Shows} =
Pr{(1,2)}+Pr{(2,1) }+Pr{ (3,4) }+Pr{ (4,3) }+Pr{ (4,4) } =
(1/16)+ (1/16)+ (1/16)+ (1/16)+ (1/16) = 5/16
Begin work on Pairs of
Dice and Random Variables
c