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

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)}

Rare?

(d4,d4)

 1/4

 1/4

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

100*(1/16) = 6.25000

No

(d5,d5)

 1/5

 1/5

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

100*(1/25) = 4.00000

No

(d8,d8)

 1/8

 1/8

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

100*(1/64) ≈ 1.56250

Borderline

(d10,d10)

   1/10

   1/10

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

100*(1/100) = 1.00000

Borderline

(d20,d20)

   1/20

   1/20

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

100*(1/400) = 0.25000

Yes

(d30,d30)

   1/30

   1/30

(1/30)*(1/30) = 1/900 ≈ 0.00111

100*(1/900) ≈ 0.11111

Yes

(d6,d4)

   1/6 

   1/4 

(1/6)*(1/4) = 1/24 ≈ 0.04167

100*(1/24) ≈ 4.16667

No

(d6,d6)

   1/6 

   1/6 

(1/6)*(1/6) = 1/36 ≈ 0.02778

100*(1/36) ≈ 2.77778

No

(d8,d10)

   1/8 

   1/10

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

100*(1/80) = 1.25000

Borderline

(d12,d12)

   1/12

   1/12

(1/12)*(1/12) = 1/144 ≈ 0.00694

100*(1/144) ≈ 0.69444

Yes

(d24,d24)

   1/24

   1/24

(1/24)*(1/24) = 1/576 ≈ 0.00174

100*(1/576) ≈ 0.17361

Yes

(d20,d30)

   1/20

   1/30

(1/20)*(1/30) = 1/600 ≈ 0.00167

100*(1/600) ≈ 0.16667

Yes

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

 

Event Probability = Pr{Event}

Sample Size = n

Expected Count = n*Pr{Event}

1/Event Probability = Minimum Sample Size

0.01

10

0.1

100

0.01

25

0.25

100

0.01

50

0.5

100

0.01

75

0.75

100

0.01

90

0.9

100

0.01

100

1

100

0.01

110

1.1

100

0.01

125

1.25

100

0.01

150

1.5

100

0.01

175

1.75

100

0.01

200

2

100

0.01

250

2.5

100

0.01

300

3

100

0.01

350

3.5

100

0.01

400

4

100

0.01

500

5

100

0.01

750

7.5

100

0.01

1000

10

100

 

Sample Size

Minimum Probability = 1/n

10

0.1

25

0.04

50

0.02

100

0.01

125

0.008

150

0.006666667

200

0.005

250

0.004

300

0.003333333

400

0.0025

500

0.002

750

0.001333333

1000

0.001

2500

0.0004

5000

0.0002

7500

0.000133333

10000

0.0001

50000

0.00002

 

 

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

Build analytic narratives for each of your case types – at a minimum, think in terms of a back-up tool if you blank out or panic on a test. Think in terms of a checklist with associated writing samples and formatting tips. Work the same case type across each test, building your expertise as you go – study one case type at a time.

The Analytic Narrative

An analytic narrative is the equivalent of a choreographic chart – things to do, how to do them and the order in which they are to be done. To build an analytic narrative for a problem, one must lay out the things required to solve the problem, in the order in which these things must be done, and how to do them.

Course Case Types and the Analytic Narrative

The testable cases in my course can be grouped into case types – groups of cases that are similar in terms of how they are solved. One can build an analytic narrative for each case type in my course, and these narratives can be very helpful in writing a good test. Understanding the case types and building analytic narratives for each case type can organize and streamline the process of preparing for a test.

The analytic narratives should form the core of your tool-sheets and can protect a student from panic during a test. Even in a panicked state, a student can follow the steps listed in a good narrative.

Contents of an Analytic Narrative

The Mathematical Part

Identify each step required to render the numerical solution of the case. Do this in both written (word) and algebra (symbols). Write this part as a step-by-step procedure.

The Writing Part

Identify each part of what is to be written, and the required or preferred formatting.