8th September 2010

Summaries

Session 1.6

 

Rare Events

The Scale of Probability

Probabilities range from 0 to 1.

Pr{Event} = 0 The event is impossible.

 

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

 

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

 

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

Pr{Event} > ฝ 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

(1st D3, 2nd 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 1st D3}*Pr{1 shows from 2nd 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

(1st D4, 2nd 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 1st D4}*Pr{1 shows from 2nd 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}

(1st D4, 2nd 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 1st D4}*Pr{1 shows from 2nd 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}

(1st D5, 2nd 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 1st D5}*Pr{1 shows from 2nd 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}

(1st D5, 2nd 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 1st D5}*Pr{1 shows from 2nd 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}

(1st D6, 2nd 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 1st D6}*Pr{1 shows from 2nd 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}

(1st D6, 2nd 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 1st D6}*Pr{1 shows from 2nd 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}

(1st D8, 2nd 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 1st D8}*Pr{1 shows from 2nd 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} 

(1st D8, 2nd 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 1st D8}*Pr{1 shows from 2nd 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} 

(1st D10, 2nd 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 1st D10}*Pr{1 shows from 2nd 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 1st D10}*Pr{1 shows from 2nd 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 1st D12}*Pr{1 shows from 2nd 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 1st D20}*Pr{1 shows from 2nd 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 1st D24}*Pr{1 shows from 2nd 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 1st D30}*Pr{1 shows from 2nd 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 1st D20}*Pr{1 shows from 2nd 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}

Pr{(1,1) shows} = Pr{1 shows from 1st D24}*Pr{1 shows from 2nd 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}

Pr{(1,1) shows} = Pr{1 shows from 1st D24}*Pr{1 shows from 2nd 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}

Pr{(1,1) shows} = Pr{1 shows from 1st D30}*Pr{1 shows from 2nd 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) @ 0.14

The 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}

Pr{(1,1) shows} = Pr{1 shows from 1st D30}*Pr{1 shows from 2nd 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

6:30

Pair

Pr{1 on 1st}

Pr{1 on 2nd}

Pr{(1,1)} = Pr{1 on 1st}*Pr{1 on 2nd}

E100 = 100*Pr{(1,1)}

observed

proportion

Rare?

(d4,d4)

 1/4

 1/4

(1/4)*(1/4) = 1/16 = 0.06250

100*(1/16) = 6.25000

6

0.0600

No

(d5,d5)

 1/5

 1/5

(1/5)*(1/5) = 1/25 = 0.04000

100*(1/25) = 4.00000

3

0.0300

No

(d8,d8)

 1/8

 1/8

(1/8)*(1/8) = 1/64 = 0.01563

100*(1/64) ≈ 1.56250

2

0.0200

Borderline

(d10,d10)

   1/10

   1/10

(1/10)*(1/10) = 1/100 = 0.01000

100*(1/100) = 1.00000

3

0.0300

Borderline

(d20,d20)

   1/20

   1/20

(1/20)*(1/20) = 1/400 = 0.00250

100*(1/400) = 0.25000

0

0.0000

Yes

(d30,d30)

   1/30

   1/30

(1/30)*(1/30) = 1/900 ≈ 0.00111

100*(1/900) ≈ 0.11111

0

0.0000

Yes

8:00

Pair

Pr{1 on 1st}

Pr{1 on 2nd}

Pr{(1,1)}

E100

observed

proportion

Rare?

(d6,d4)

   1/6 

   1/4 

(1/6)*(1/4) = 1/24 ≈ 0.04167

100*(1/24) ≈ 4.16667

3

0.0300

No

(d6,d6)

   1/6 

   1/6 

(1/6)*(1/6) = 1/36 ≈ 0.02778

100*(1/36) ≈ 2.77778

5

0.0500

No

(d8,d10)

   1/8 

   1/10

(1/8)*(1/10) = 1/80 = 0.01250

100*(1/80) = 1.25000

1

0.0100

Borderline

(d12,d12)

   1/12

   1/12

(1/12)*(1/12) = 1/144 ≈ 0.00694

100*(1/144) ≈ 0.69444

0

0.0000

Yes

(d24,d24)

   1/24

   1/24

(1/24)*(1/24) = 1/576 ≈ 0.00174

100*(1/576) ≈ 0.17361

0

0.0000

Yes

(d20,d30)

   1/20

   1/30

(1/20)*(1/30) = 1/600 ≈ 0.00167

100*(1/600) ≈ 0.16667

0

0.0000

Yes

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 Probability Case Types

ึLong Run Argument/Perfect Samples – should be finished

ึProbability Rules – should be finished, except for the conditional probability bits

ึColor Slot Machine – should be finished, except for the conditional probability bits

ึPairs of Dice – should be nearly finished, except for the conditional probability bits

ึRandom Variables – should be nearly finished, except for the conditional probability bits

Next  Case Types: Conditional Probability