1. Section 5.2 - Using Simulation to Estimate Probabilities
P12. A catastrophic accident is one that involves severe skull or spinal damage. The National Center for Catastrophic Sports Injury Research reports that over the last 21 years, there have been 101 catastrophic accidents among female high school and college athletes. Fifty-five of these resulted from cheerleading.
Suppose you want to study catastrophic accidents in more detail, and you take a random sample, without replacement, of 8 of these 101 accidents. Estimate the probability that at least half of your eight sampled accidents resulted from cheerleading.
Start at the beginning of row 17 of Table D on page 828, and add ten runs to the frequency table.

2. Section 5.2 - Using Simulation to Estimate Probabilities
P12. Of the 101 catastrophic accidents among female high school and college athletes, 55 of resulted from cheerleading.
Take a random sample, without replacement, of 8 of these 101 accidents. Estimate the probability that at least half of your eight sampled accidents resulted from cheerleading.
Assumptions:
You have a random sample (without replacement) of size n = 8 from a population of 101 accidents, of which 55 resulted from cheerleading. (Each sample of size n = 8 is equally likely.)

3. Section 5.2 - Using Simulation to Estimate Probabilities
P12. Model:
There are 1000 three-digit triples and you wish to represent 101 accidents, so it is convenient to assign 9 three-digit numbers to each accident, as follows. (If you assign the numbers , you will have to reject 90% of the numbers selected. This method rejects only 91 or 9.1%)
Triple-Digit Numbers
Accident Number 1 2 … 99
100
010, 020, 030, 040, 050, 060, 070, 080, 090
101

4. Section 5.2 - Using Simulation to Estimate Probabilities
P12. Model:
Assign 9 three-digit numbers to each accident, as follows. Ignore all triples not on the table. Accidents numbered from 1 through 55 will be considered as resulting from cheerleading. Stop a run when you have a sample of 8 accidents (no repeats). Repeat until you have 10 runs.
Triple-Digit Numbers
Accident Number 1 2 … 99
100
010, 020, 030, 040, 050, 060, 070, 080, 090
101

5. Section 5.2 - Using Simulation to Estimate Probabilities
P12. Repetition:
Starting on line 17 of Table D, divide the digits into triples. Triples to be ignored are crossed out and the remaining ones are grouped into sequences representing samples of 8 accidents, making sure there are no repeated accidents in any one run.
Run 1 = 4 from cheerleading
Triple:
801
243
563
517
727
080
154
531
Accident: 80 24 56 51 72
101 15 53
Result: no
yes
Run 2 = 6 from cheerleading
822
374
211
157
825
314
385
537
637 82 37 21 31 38 63 x

6. Section 5.2 - Using Simulation to Estimate Probabilities
P12. Repetition:
Run 3 = 5 from cheerleading
Triple:
435
099
817
774
027
721
443
236
Accident: 43 9 81 77 2 72 44 23
Result:
yes no
Run 4 = 6 from cheerleading
002
104
552
164
237
962
860
265
569
100 10 55 16 96 x 26 56
Run 5: 6 from cheerleading
916
268
036
625
229
148
369
368
720
376 91 3 62 22 14 36 37

7. Section 5.2 - Using Simulation to Estimate Probabilities
P12. Repetition:
Run 6 = 4 from cheerleading
Triple:
621
139
909
440
056
418
098
932
050
Accident: 62 13 90 x 5 41 9 93
101
Result: no
yes
Run 7 = 3 from cheerleading
514
225
685
144
642
756
788
962 51 22 68 14 64 75 78 96
Run 8: 4 from cheerleading
977
882
254
382
145
989
149
914
523 97 88 25 38 98 91 52

8. Section 5.2 - Using Simulation to Estimate Probabilities
P12. Repetition:
Run 9 = 3 from cheerleading
Triple:
684
792
768
646
162
835
549
475
Accident: 68 79 76 64 16 83 54 47
Result: no
yes
Run 10 = 2 from cheerleading
089
923
370
892
004
880
336
945
982
694 8 92 x 89
100 33 94 98 69
Summary: 4,6,5,5,6,4,3,4,3,2

9. Section 5.2 - Using Simulation to Estimate Probabilities
P12. Repetition: Update Display 5.22
Accidents from Cheerleading
Frequency 1 10 2 81 3
187 4
262 5
252 6
162 7 43 8
Total
1000

10. Section 5.2 - Using Simulation to Estimate Probabilities
P12. Conclusion:
Out of the 1000 runs, 722 samples had at least half (four or more) of the accidents from cheerleading.
The estimated probability that a random sample of eight catastrophic accidents would have at least half of the accidents from cheerleading is about
This is close to the theoretical probability of computed using the hypergeometric distribution.
See Fathom file SIA Ch 5 P12.ftm

11. Section 5.2 - Using Simulation to Estimate Probabilities
P13. The winner of the World Series is the first team to win four games. That means the series can be over in four games or can go as many as seven games. Suppose the two teams playing are evenly matched.
Estimate the probability that the World Series will go seven games before there is a winner.
Start at the beginning of row 9 of Table D on page 828, and add your ten runs to the frequency table.

12. Section 5.2 - Using Simulation to Estimate Probabilities
P13. The winner of the World Series is the first team to win four games. Suppose the two teams playing are evenly matched. Estimate the probability that the World Series will go seven games before there is a winner.
Assumptions:
The probability of a given team winning a particular game is 50% and that the results of the games are independent of each other. (This is probably not a realistic assumption!)

13. Section 5.2 - Using Simulation to Estimate Probabilities
P13. Model:
Assign digits 1-5 to team A winning and 6-9, 0 to team B winning. Select digits from the table, allowing repeats, until one of the teams has four wins.
Record the total number of digits needed to get four wins for one of the teams.
Digits
Winning Team
1 - 5 A
6 - 9 , 0 B

14. Section 5.2 - Using Simulation to Estimate Probabilities
P13. Repetition:
Starting on line 09 of Table D:
Run 1 2 3 4 5
Digits
635733
2135
05325
470489
05535
Team
BAABAA
AAAA
BAAAA
ABBABB
Games 6 7 8 9 10
68287
098349
12562
473796
BAABABA
BBABB
BBBAAB
AAABA
ABABBB

15. Section 5.2 - Using Simulation to Estimate Probabilities
P13. Repetition: Update Display 5.23
Games in Series
Frequency 4
611 5
1307 6
1545 7
1537
Total
5000

16. Section 5.2 - Using Simulation to Estimate Probabilities
P13. Conclusion:
Out of the 5000 runs, 1537 series went the full seven games, so the estimated probability of a World Series of two evenly matched teams going seven games is 1537 / 5000, or
The theoretical probability is When the teams are not evenly matched, the probability is less.
In the 66 World Series between 1940 and 2006, the World Series went the full seven games 27 times out of 66, or
This is not statistically significant, but close.
One reasonable explanation is that the probability of winning changes from game to game.

17. Section 5.2 - Using Simulation to Estimate Probabilities
E15. About 10% of high school girls report that they rarely or never wear a seat belt while riding in motor vehicles. Suppose you randomly sample four high school girls.
Estimate the probability that no more than one of the girls says that she rarely or never wears a seat belt.
Start at the beginning of row 36 of Table D on page 828, and add your ten results to the frequency table.

18. Section 5.2 - Using Simulation to Estimate Probabilities
E15. About 10% of high school girls report that they rarely or never wear a seat belt while riding in motor vehicles. Suppose you randomly sample four high school girls.
Estimate the probability that no more than one of the girls says that she rarely or never wears a seat belt.
Assumptions:
The probability that a randomly selected girl reports that she rarely or never wears a seat belt is 10%, and that the girls are selected independently of each other.

19. Section 5.2 - Using Simulation to Estimate Probabilities
E15. Model:
Assign the digits 0 to 9 as follows:
Reports that she rarely or never wears a seat belt: 0
Does not report that she rarely or never wears a seat belt: 1-9

20. Section 5.2 - Using Simulation to Estimate Probabilities
E15. Repetition:
Begin with row 36 of table D, separated into groups of 4:
Run: 1 2 3 4 5 6 7 8 9 10
Digits:
3217
9005
9787
3792
5241
0556
7070
0786
7431
7157
Result:

21. Section 5.2 - Using Simulation to Estimate Probabilities
E15. Repetition:
Add the results to the table in Display 5.24:
Number of Girls Who Rarely or Never Wear a Seat Belt
Frequency
6,641 1
2,863 2
464 3 32 4
Total
10,000

22. Section 5.2 - Using Simulation to Estimate Probabilities
E15. Conclusion:
Out of the 10,000 runs, = 9504 had no more than one 0.
The estimated probability that a random sample of four girls contains no more than one that says she rarely or never wears a seat belt is
The theoretical probability is

23. Section 5.2 - Using Simulation to Estimate Probabilities
E18. A Harris poll estimated that 25% of U.S. residents believe in astrology. Suppose you would like to interview a person who believes in astrology.
Estimate the probability that you will have to ask four or more U.S. residents to find one who believes in astrology.
Start at the beginning of row 15 of Table D on page 828, and add your ten results to the frequency table.

24. Section 5.2 - Using Simulation to Estimate Probabilities
E18. A Harris poll estimated that 25% of U.S. residents believe in astrology. Suppose you would like to interview a person who believes in astrology.
Estimate the probability that you will have to ask four or more U.S. residents to find one who believes in astrology.
Assumptions:
The probability that a randomly selected person believes in astrology is 25%, and that each person was selected randomly and independently from the population of U.S. residents.

25. Section 5.2 - Using Simulation to Estimate Probabilities
E18. Model:
Assign two digit-numbers as follows:
Believes in astrology:
Does not believe in astrology: , 00

26. Section 5.2 - Using Simulation to Estimate Probabilities
E18. Repetition:
Begin with row 15 of table D:
Run 1 2 3
Digits
68 23
Belief:
N N N N N N N N N N
N N N N N N N N Y
N Y
N N N Y
People 19 4

27. Section 5.2 - Using Simulation to Estimate Probabilities
E18. Repetition:
Begin with row 15 of table D:
Run 4 5 6 7 8
Digits
Belief:
N N Y
N N N N Y
People 3
Run 9 10
Digits
Belief:
N N N N Y
People 5

28. Section 5.2 - Using Simulation to Estimate Probabilities
E18. Repetition:
Add the results to the table in Display 5.24:
Number of People Asked
Frequency 1
437 2 3 4
210 +1 5
173 +3 … 19
4 + 1
Total
2,000

29. Section 5.2 - Using Simulation to Estimate Probabilities
E18. Conclusion:
Out of the 2,000 runs, ( ) = 868 had four or more.
The estimated probability that you would have to interview four or more U.S. residents before getting a person who believes in astrology is 868 / 2000 or 0.434
The theoretical probability is
(You will learn how to compute this in Section 6.3)

30. Section 5.2 - Using Simulation to Estimate Probabilities
E19. The probability that a baby is a girl is about Suppose a large number of couples each plan to have babies until they have a girl.
Estimate the average number of babies per couple.
Start at the beginning of row 9 of Table D on page 828, and add your ten results to the frequency table in Display 5.28, which gives the results of 1990 runs.

31. Section 5.2 - Using Simulation to Estimate Probabilities
E19. The probability that a baby is a girl is about Suppose a large number of couples each plan to have babies until they have a girl.
Estimate the average number of babies per couple.
Assumptions:
Each baby has a 0.49 chance of being a girl.
The gender of each child is independent of the other children’s gender in that family.

32. Section 5.2 - Using Simulation to Estimate Probabilities
E19. Model:
Assign pairs of digits as follows:
= girl baby
, 00 = boy baby
A single run consists of selecting pairs of digits, allowing repeats, until a pair in the range is selected.
Record the number of pairs needed.

33. Section 5.2 - Using Simulation to Estimate Probabilities
E19. Repetition:
Start with Row 9 of Table D. Perform 10 repetitions.
63 / 57 / 33 // 21 // 35 // 05 // 32 // 54 / 70 / 48 //
90 / 55 / 35 // 75 / 48 // 28 // 46 //
The numbers needed are:
3, 1, 1, 1, 1, 3, 3, 2, 1, 1

34. Section 5.2 - Using Simulation to Estimate Probabilities
E19. Repetition:
Add the results of our ten runs to the frequency table:
Babies
Frequency 1
= 947 8 11 2
= 481 9 4 3
= 268 10
156 5 60 12 6 51 13 7 17
Total
= 2000

35. Section 5.2 - Using Simulation to Estimate Probabilities
E19. Conclusion:
Apply the method of calculating a mean from a frequency table: Enter the data into L1, L2; Run 1-Var Stats.
Babies
Frequency 1
= 947 8 11 2
= 481 9 4 3
= 268 10
156 5 60 12 6 51 13 7 17
Total
= 2000

36. Section 5.2 - Using Simulation to Estimate Probabilities
E19. Conclusion:
Apply the method of calculating a mean from a frequency table: Enter the data into L1, L2; Run 1-Var Stats.
The estimated average number of babies for a family that keeps having babies until they have a girl is about children.
The theoretical mean is about 2.04.

37. Section 5.2 - Using Simulation to Estimate Probabilities
E20. Boxes of cereal often have small prizes in them. Suppose each box of one type of cereal contains one of four different small cars.
Estimate the average number of boxes a parent will have to buy until his or her child gets all four cars.
Start at the beginning of row 49 of Table D on page 828, and add your ten results to the frequency table.

38. Section 5.2 - Using Simulation to Estimate Probabilities
E20. Boxes of cereal often have small prizes in them. Suppose each box of one type of cereal contains one of four different small cars.
Estimate the average number of boxes a parent will have to buy until his or her child gets all four cars.
Assumptions:
The different types of cars are equally and randomly distributed in the boxes of cereal so the probability if getting a particular type of car is always Each box of cereal is selected independently.

39. Section 5.2 - Using Simulation to Estimate Probabilities
E20. Model:
Assign two digit-numbers as follows:
Car 1:
Car 2:
Car 3:
Car 4: , 00

40. Section 5.2 - Using Simulation to Estimate Probabilities
E20. Repetition:
Begin with row 49 of table D:
Run 1
Digits
Cars
Boxes 13
Run 2
Digits
Cars
Boxes 11

41. Section 5.2 - Using Simulation to Estimate Probabilities
E20. Conclusion:
Boxes
Frequency 4
887 5
1395 6
1488 7
1347 8
1136 9
903 10
679 11
491 12
369 13
331
Boxes
Frequency 14
263 15
167 16
137 17
103 18 81 19 67 20 44 21 29 22 23
Boxes
Frequency 24 7 25 12 26 5
27+
Total
10,000

42. Section 5.2 - Using Simulation to Estimate Probabilities
E20. Conclusion:
We were asked to estimate a mean, not a probability.
Count 27+ as 27.

43. Section 5.2 - Using Simulation to Estimate Probabilities
E20. Conclusion:
We were asked to estimate a mean, not a probability.
The estimated mean number of boxes a parent must buy to get all four cars is

44. Section 5.2 - Using Simulation to Estimate Probabilities
E21. A group of five friends have backpacks that all look alike. They toss their backpacks on the ground and later pick up a backpack at random.
Estimate the probability that everyone gets his of her own backpack.
Start at the beginning of row 31 of Table D on page 828.

45. Section 5.2 - Using Simulation to Estimate Probabilities
E21. A group of five friends have backpacks that all look alike. They toss their backpacks on the ground and later pick up a backpack at random.
Estimate the probability that everyone gets his of her own backpack.
Assumptions:
Each backpack is equally likely to be picked up by each person. The backpacks are selected independently of each other.

46. Section 5.2 - Using Simulation to Estimate Probabilities
E21. Model:
Backpack 1 belongs to person 1, backpack 2 belongs to person 2, etc.
Assign digits to represent backpacks: digits 1 and 2 = backpack 1; digits 3 and 4 = backpack 2; digits 5 and 6 = backpack 3; digits 7 and 8 = backpack 4; digits 9 and 0 = backpack 5.
Select 4 digits and ignore repeats.
Record the number of correct backpacks.

47. Section 5.2 - Using Simulation to Estimate Probabilities
E21. Repetition:
Begin with row 31 of table D. Perform 20 repetitions.
/ / / / /
/ / / / /
/ / 6928 / / 7295 / 2935 /
9631 / 5307 / 2689 / 80935

48. Section 5.2 - Using Simulation to Estimate Probabilities
E21. Repetition:
Run
Digits
Bags
Correct 1
52xxx31(4) 2
49475
25x43(1) 3
246338
123xx4(5) 5 4
24458
12x34(5)
31xx5xxxxxxx4(2) 6
52x3xxxx4(1) 7
124720
1x24x5(3) 8
054997
532xx4(1) 9
3x2xx5x1(4) 10
88159
4x135(2)

49. Section 5.2 - Using Simulation to Estimate Probabilities
E21. Repetition:
Run
Digits
Bags
Correct 11
531xxx2(4) 12
89654
453x2(1) 1 13
6928
3514(2) 14
125xxxx4(3) 3 15
7295
4153(2) 16
2935
1523(4) 17
9631
5321(4) 18
5307
3254(1) 2 19
2689
1345(2) 20
80935
45x23(1)

50. Section 5.2 - Using Simulation to Estimate Probabilities
E21. Repetition:
Correct
Frequency
Probability 5
5 / 20 = 0.25 1 8
8 / 20 = 0.40 2 3
3 / 20 = 0.15
2 / 20 = 0.10 4
0 / 20 = 0.00
Total 20
1.00

51. Section 5.2 - Using Simulation to Estimate Probabilities
E21. Conclusion:
The estimated probability that each person gets his or her own backpack is 0.10.
The theoretical probability is

52. Section 5.2 - Using Simulation to Estimate Probabilities
E23. There are six different car keys in a drawer, including yours. Suppose you grab one key at a time until you get your car key.
Estimate the probability that you get your car key on the second try.
Start at the beginning of row 13 of Table D on page 828.

53. Section 5.2 - Using Simulation to Estimate Probabilities
E23. There are six different car keys in a drawer, including yours. Suppose you grab one key at a time until you get your car key.
Estimate the probability that you get your car key on the second try.
Assumptions:
You select a key at random and each key is equally likely to be picked on any one grab

54. Section 5.2 - Using Simulation to Estimate Probabilities
E23. Model:
Assign the digit 1 to your key and the digits to the other keys. Ignore the digits 7, 8, 9, 0.
A single run consists of selecting two digits from (ignore repeats, ignore 7, 8, 9, 0).
There is no need to continue selecting after the second valid digit.

55. Section 5.2 - Using Simulation to Estimate Probabilities
E23. Repetition:
Begin with row 13 of table D:
834 / 52 / 9963 / 406 / / 13 / 746 / /
754 / 061 / / / 854 / / /
013 / / 1903 / 64 / 764 /

56. Section 5.2 - Using Simulation to Estimate Probabilities
E23. Repetition:
834 / 52 / 9963 / 406 / / 13 / 746 / /
754 / 061 / / / 854 / / /
013 / / 1903 / 64 / 764 /
Second Key Yours?
Frequency
Yes 2 No 18
Total 20

57. Section 5.2 - Using Simulation to Estimate Probabilities
E23. Conclusion:
This is a small simulation of only 20 runs.
The estimated probability that if you pull one key at a time out of a drawer, you will get your key on the second draw is 2 / 20, or The theoretical probability is 1/6. Why?

58. Section 5.2 - Using Simulation to Estimate Probabilities
E23. Conclusion:
The estimated probability that if you pull one key at a time out of a drawer, you will get your key on the second draw is 2 / 20, or The theoretical probability is 1/6. Why?

59. Section 5.2 - Using Simulation to Estimate Probabilities
E24. A deck of cards contains 13 hearts. Suppose you draw cards one at a time, without replacement.
Estimate the probability that it takes you four cards or more to draw the first heart.
Start at the beginning of row 28 of Table D on page 828.

60. Section 5.2 - Using Simulation to Estimate Probabilities
E24. A deck of cards contains 13 hearts. Suppose you draw cards one at a time, without replacement.
Estimate the probability that it takes you four cards or more to draw the first heart.
Assumptions:
Each card is equally likely to be drawn and are selected independently.

61. Section 5.2 - Using Simulation to Estimate Probabilities
E24. Model:
Assign the digits to the cards in the deck.
Let the digits represent the hearts.
Ignore all other digits: , 00.
A single run consists of selecting pairs of digits without repeats until you get a heart ( )
Record the number of pairs needed to get a heart.

62. Section 5.2 - Using Simulation to Estimate Probabilities
E23. Repetition:
Begin with row 28 of table D:
Run 1:

63. Section 5.2 - Using Simulation to Estimate Probabilities
E24. Repetition:
Run
Draws 1 6 2 4 3 5 7 12 8 9 10
Run
Draws 11 2 12 7 13 4 14 3 15 16 1 17 18 19 20
Draws
Frequency
1, 2, 3 10 4+
Total 20

64. Section 5.2 - Using Simulation to Estimate Probabilities
E24. Conclusion:
This is a small simulation of only 20 runs.
The estimated probability that it takes four or more draws from a standard deck to get a heart is 10 / 20, or 0.50.
The theoretical probability is Why?

65. Section 5.2 - Using Simulation to Estimate Probabilities
E24. Conclusion:
This is a small simulation of only 20 runs.
The estimated probability that it takes four or more draws from a standard deck to get a heart is 10 / 20, or 0.50.
The theoretical probability is Why?

66. Section 5.2 - Using Simulation to Estimate Probabilities
E24. Conclusion:
This is a small simulation of only 20 runs.
The estimated probability that it takes four or more draws from a standard deck to get a heart is 10 / 20, or 0.50.
The theoretical probability is Why?

67. Section 5.2 - Using Simulation to Estimate Probabilities
E26. This question appeared in the “Ask Marilyn” column:
I work at a waste-treatment plant, and we do assessments of the time-to-failure and time-to-repair of the equipment, then input those figures into a computer model to make plans. But when I need to explain the process to people in other departments, I find it difficult.
Say a component has two failure modes. One occurs every 5 years, and the other occurs every 10 years. People usually say that the time to failure is 7.5 years, but this is incorrect. It’s between 3 and 4 years. Do you know of a way to explain this that people will accept?

68. Section 5.2 - Using Simulation to Estimate Probabilities
E26. Say a component has two failure modes. One occurs every 5 years, and the other occurs every 10 years. People usually say that the time to failure is 7.5 years, but this is incorrect. It’s between 3 and 4 years. Do you know of a way to explain this that people will accept?
Use simulation to estimate the expected time to failure for this component. Start at the beginning of row 40 of Table D.

69. Section 5.2 - Using Simulation to Estimate Probabilities
E26. Say a component has two failure modes. One occurs every 5 years, and the other occurs every 10 years.
Use simulation to estimate the expected time to failure for this component.
Assumptions:
The probability of a mode 1 failure in a given year is 20%. The probability of a mode 2 failure in a given year is 10%. These failures occur independently of each other.

70. Section 5.2 - Using Simulation to Estimate Probabilities
E26. Model:
Consider pairs of digits.
The first digit represents mode 1 status, with 1 or 2 representing failure.
The second digit represents mode 2 status, with 1 representing failure.
Thus, a failure is any one of the 28 pairs:
10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,
29,01,31,41,51,61,71,81,91 (Why 28 pairs?)
A single run consists of selecting pairs of digits until you get a failure. Perform a large number of runs, and compute the mean number of years until failure

71. Section 5.2 - Using Simulation to Estimate Probabilities
E26. Repetition:
Begin with row 40 of table D:
Run 1: 4 years
Run 2: 3 years
Run 3: 1 year
Run 4: 2 years
8 1
0851 81
08 51
ok ok ok fail
ok ok fail
fail
ok fail

72. Section 5.2 - Using Simulation to Estimate Probabilities
E26. Repetition:
Run
Years 1 4 2 3 5 6 7 8 9 10
Run
Years 11 16 12 1 13 3 14 15 2 17 7 18 4 19 9 20 5

73. Section 5.2 - Using Simulation to Estimate Probabilities
E26. Conclusion:
This is a small simulation of only 20 runs.
The estimated mean number of years until failure is 83/20 = 4.15 years.
The theoretical expected time to failure is 3.57.
P(failure) = 1 - P(no failures) = 1 - (0.8)(0.9) = 0.28 (failures/yr)
Expected time = 1 / 0.28 = 3.57 (yrs/failure)