]> Answers to Selected Exercises

Answers to Selected Exercises

2. Probability Spaces

  1. Events and Random Variables
  2. Probability Spaces
  3. Conditional Probability
  4. Independence
  5. Convergence

2. Events and Random Variables

2.18.

  1. Y x 1 x 2 x n x 1 x 2 x n . The set of possible values is 0 1 n
  2. 11100 11010 11001 10110 10101 10011 01110 01101 01011 00111

2.20.

  1. Y x 1 x 2 x n x 1 x 2 x n . The set of possible values is n n 1 n k
  2. U x 1 x 2 x n x 1 x 2 x n . The set of possible values is 1 2 k
  3. V x 1 x 2 x n x 1 x 2 x n . The set of possible values is 1 2 k
  4. u v 1 2 k 2 u v

2.21.

  1. A 1 1 1 2 1 3 1 4 1 5 1 6
  2. B 1 6 2 5 3 4 4 3 5 2 6 1
  3. A B 1 1 1 2 1 3 1 4 1 5 1 6 2 5 3 4 4 3 5 2 6 1
  4. A B 1 6
  5. A B A B 2 1 2 2 2 3 2 4 2 6 3 1 3 2 3 3 3 5 3 6 4 1 4 2 4 4 4 5 4 6 5 1 5 3 5 4 5 5 5 6 6 2 6 3 6 4 6 5 6 6

2.23.

  1. X 1 3 X 2 4 1 5 2 5 1 6 2 6
  2. Y 7 1 6 2 5 3 4 4 3 5 2 2 5 6 1
  3. U V 1 1 2 2 3 3 4 4 5 5 6 6

2.25. Let D 5 1 4 2 3 3 2 4 1 , D 7 1 6 2 5 3 4 4 3 5 2 6 1 , D D 5 D 7 , and C 1 2 3 4 5 6 2 D .

  1. S D C D C 2 D , A D 5 C D 5 C 2 D 5
  2. S D , A D 5

2.26.

  1. S 1 2 3 4 5 6 3
  2. W x 1 x 2 x 3 x 1 6 x 2 6 x 3 6 1

2.27. Let 1 denote heads and 0 tails for a coin toss.

  1. n 1 6 0 1 n , S 126
  2. N x 1 x 2 x n n for x 1 x 2 x n S
  3. Y x 1 x 2 x n i 1 n x i for x 1 x 2 x n S
  4. Y 2 11 011 101 110 0011 0101 0110 1001 1010 1100 00011 00101 00110 01001 01010 01100 10001 10010 10100 11000 000011 000101 000110 001001 001010 001100 010001 010010 010100 011000 100001 100010 100100 101000 110000

2.29. For the coin, let 1 denote heads and 0 tails.

  1. S 0 1 1 2 3 4 5 6 , S 12
  2. X i j i for i j S
  3. Y i j j for i j S
  4. Y 4 0 1 4 5 6

2.31.

  1. 311875200, 2598960
  2. 3954242643911239680000, 635013559600

2.32.

  1. Q q q q q
  2. H 1 2 10 j q k
  3. Q H 1 2 10 j q k q q q
  4. Q H q
  5. Q H q q q

2.34.

  1. The set of possible values of V is 0 1 37
  2. V 0 2310789600

2.36.

  1. A 3744
  2. B 624
  3. A 5148

2.38.

  1. S 12 12 2
  2. A r 12 12 r 2
  3. A x y S x r 12 x 12 r y r 12 y 12 r
  4. Z x y x 2 y 2 for x y S
  5. X Y x y S x y
  6. Z 12 x y S x 2 y 2 14

2.42.

  1. 254251200
  2. 2118760
  3. 658008, 913900, 444600, 936000, 8400, 252

2.47.

  1. U 3 1 X 1 + X 2 + X 3 - X 1 X 2 - X 1 X 3 - X 2 X 3 + X 1 X 2 X 3
  2. U 3 2 X 1 X 2 X 1 X 3 X 2 X 3 2 X 1 X 2 X 3
  3. U 3 3 X 1 X 2 X 3

2.48. Y X 3 X 1 X 2 X 1 X 2 X 4 X 5 X 4 X 5 1 X 3 X 1 X 4 X 2 X 5 X 1 X 2 X 4 X 5

2.55. For gender, let 0 denote female and 1 male. For species, let 1 denote tredecula, 2 tredecim, and 3 tredecassini.

  1. S 0 4 0 1 1 2 3
  2. F x 1 x 2 x 3 x 4 y z S y 0
  3. S 104 where S is given in (a).

2.56.

  1. S 6 0
  2. A n 1 n 2 n 3 n 4 n 5 n 6 w S n 1 n 2 n 3 n 4 n 5 n 6 57
  3. S 30 where S is given in (a).

3. Probability Measure

3.29.

  1. A occurs but not B . A B 730
  2. A or B occurs. A B 2960
  3. One of the events does not occur. A B 910
  4. Neither event occurs. A B 3160
  5. Either A occurs or B does not occur. A B 1720

3.30.

  1. A B C 0.67
  2. A B C 0.33
  3. A B C A B C A B C 0.45
  4. A B C A B C A B C 0.21

3.31.

  1. A 14
  2. B 13
  3. A B 12
  4. A B 1112
  5. A B 12

3.32.

  1. B 12
  2. A B 15
  3. B A 310
  4. A B 45
  5. A B 310

3.33.

  1. Probabilities of Y
    k 0 1 2 3 4 5
    Y k 132 532 1032 1032 532 132

3.34.

  1. A 12
  2. B 38
  3. A B 14
  4. A B 58
  5. A B 34
  6. A B 38
  7. A B 78

3.37.

  1. A X 1 3
  2. B X 1 X 2 6
  3. A 13
  4. B 536
  5. A B 236
  6. A B 512
  7. B A 112

3.39.

  1. Y y 6 y 7 36 for y 2 3 12
  2. U u 13 2 u 36 for u 1 2 6
  3. V v 2 v 1 36 for v 1 2 6
  4. U u V v 236 u v 136 u v

3.40. Let D 5 1 4 2 3 3 2 4 1 , D 7 1 6 2 5 3 4 4 3 5 2 6 1 , D D 5 D 7 , and C 1 2 2 4 5 6 2 D .

  1. S D C D C 2 D
  2. A D 5 C D 5 C 2 D 5
  3. A 25
  4. S D
  5. A D 5
  6. A 25

3.42.

  1. H 1 14
  2. H 1 H 2 117
  3. H 2 H 1 1368
  4. H 2 14
  5. H 1 H 2 1534

3.44.

  1. 37442598960 0.001441
  2. 6242598960 0.000240
  3. 51482598960 0.001981

3.46. 347373600635013559600 0.000547

3.47.

  1. 151519319380635013559600 0.2386
  2. 47079732700635013559600 0.0741
  3. 11404407300635013559600 0.0179

3.48.

  1. 19134962598960 0.7363
  2. 32427298180635013559600 0.0511

3.49.

  1. S 12 12 2
  2. Since the coin is tossed "randomly," no region of S should be preferred over any other.
  3. r 12 X 12 r r 12 Y 12 r
  4. A 1 2 r 2
  5. A 1 1 2 r 2
  6. Z 12 4

3.53.

Probabilities of Y
k 0 1 2 3 4 5
Y k 258423751 807523751 9502639 380023751 1003393 21131

3.55. Let U denote the urn (as a set of 12 balls).

  1. S is the set of all subsets (combinations) of size 3 chosen from U .
  2. A 344
  3. B 311

3.56. Again let U denote the urn (as a set of 12 balls).

  1. S U 3
  2. A 18
  3. B 524

3.57

  1. A 1 , B 0 , C 0
  2. A 0 , B 0 , C 1
  3. A 14 , B 12 , C 14
  4. A 0 , B 1 , C 0
  5. A 12 , B 12 , C 0
  6. A 0 , B 12 , C 12

3.58

  1. B 0 , C 0 , D 0
  2. B 12 , C 0 , D 12
  3. B 0 , C 12 , D 0
  4. B 12 , C 12 , D 12
  5. B 1 , C 12 , D 0
  6. B 1 , C 0 , D 1

3.59

  1. 3
  2. 2 4

3.60

  1. 1 52 1
  2. 1724 1

3.63

  1. 0.6333333333
  2. 0.6321205357
  3. 0.6321205588

3.64.

  1. R 1330
  2. T 1930
  3. W 930
  4. R T 930
  5. T W 1130

3.65.

  1. W 37104
  2. F 59104
  3. T 44104
  4. W F 34104
  5. W T F 85104

4. Conditional Probability

4.7.

  1. A B 25
  2. B A 310
  3. A B 35
  4. B A 710
  5. A B 3145

4.8.

  1. A B C 14
  2. A B C 712
  3. A B C 512

4.9.

  1. A B 14
  2. A B 712
  3. B A 34
  4. B A 12
  5. A and B are positively correlated.

4.10. For a person chosen at random from the population, let S denote the event that the person smokes and D the event that the person has the disease.

  1. D S 0.036
  2. S D 0.45
  3. S and D are positively correlated.

4.11.

  1. X 30 23
  2. X 45 X 30 12
  3. Given X 30 , X is uniformly distributed on 30 60

4.12.

  1. X 1 3 16 , Y 5 19 , X 1 3 Y 5 14 , Y 5 X 1 3 16 . The events are positively correlated.
  2. X 1 3 16 , Y 7 16 , X 1 3 Y 7 16 , Y 7 X 1 3 16 . The events are independent.
  3. X 1 2 16 , Y 5 19 , X 1 2 Y 5 14 , Y 5 X 1 2 16 . The events are positively correlated.
  4. X 1 3 16 , X 1 2 16 , X 1 3 X 1 2 0 , X 1 2 X 1 3 0 . The events are negatively correlated.

4.14. The conditional distribution of X 1 X 2 given Y 7 is uniform on 1 6 2 5 3 4 4 3 5 2 6 1

4.15. Let X denote the die score and H the event that all coin tosses result in heads.

  1. H 21128
  2. X i H 6463 1 2 i ,  i 1 2 3 4 5 6

4.17. Let V denote the probability of heads for the randomly selected coin, and H the event that the coin lands heads.

  1. H 4172
  2. V p H 1541 p 12 841 p 13 1841 p 1

4.18. Let X denote the die score and H the even that the coin lands heads.

  1. X i 524 i 1 6 748 i 2 3 4 5
  2. H X 4 37 ,  H X 4 47

4.20.

  1. Q 1 113 , H 1 14 , Q 1 H 1 113 , H 1 Q 1 14 , independent.
  2. Q 1 113 , Q 2 113 , Q 1 Q 2 351 , Q 2 Q 1 351 , negatively correlated.
  3. Q 2 113 , H 2 14 , Q 2 H 2 113 , H 2 Q 2 14 , independent..
  4. Q 1 113 , H 2 14 , Q 1 H 2 113 , H 2 Q 1 14 , independent.

4.22. Let H i denote the event that card i is a heart and S i the event that card i is a spade.

  1. H 1 H 2 H 3 11850
  2. H 1 H 2 S 3 13850
  3. H 1 S 2 H 3 13850

4.24.

  1. X 0 X Y 34
  2. Given X Y r 12 12 r 2 ,  X Y is uniformly distributed on this set.

4.26. Let R denote the number of reds and W the weight. R 10 W 48 1023

4.27. Let M denote the event that a cicada is male, U the event that the cicada is treducla, and W the body weight.

  1. W 0.25 M 245
  2. W 0.25 U 744

4.28. Let X denote the production line of the selected item, and D the event that the item is defective.

  1. D 0.037
  2. X i D 0.541 i 1 0.405 i 2 0.054 i 3

4.29.

  1. 78
  2. 17

4.30.

  1. 2324
  2. 323

4.31.

  1. 5.55% of the population is colorblind.
  2. 90.9% of colorblind persons are male.

4.32.

  1. 56
  2. 16
  3. 15

4.33. Let G denote the event that the ball is green and U 1 the event that urn 1 is chosen.

  1. G 920
  2. U 1 G 23

4.34. Let G i denote the event that the ball from urn i is green.

  1. G 2 925
  2. G 1 G 2 23

4.35. Let R i denote the event that the ball i is red and G i the event that ball i is green.

  1. R 1 R 2 G 3 a b a k a b a b k a b 2 k
  2. R 1 G 2 R 3 a b a k a b a b k a b 2 k
  3. G 1 R 2 R 3 a b a k a b a b k a b 2 k
  4. R 2 a a b
  5. R 1 R 2 a k a b k

4.36. Let R i denote the event that the ball i is red and G i the event that ball i is green.

  1. R 1 R 2 G 3 955
  2. R 1 G 2 R 3 744
  3. G 1 R 2 R 3 49330
  4. R 2 3255
  5. R 1 R 2 916

4.39. 0.905.

4.40. 0.949.

4.41. 0.268.

4.42. 0.9098.

5. Independence

5.8

  1. A , B , C are independent if and only if
    1. A B A B
    2. A C A C
    3. B C B C
    4. A B C A B C
  2. A , B , C , D are independent if and only if
    1. A B A B
    2. A C A C
    3. A D A D
    4. B C B C
    5. B D B D
    6. C D C D
    7. A B C A B C
    8. A B D A B D
    9. A C D A C D
    10. B C D B C D
    11. A B C D A B C D

5.20.

  1. A B C 0.93
  2. A B C 0.07
  3. A B C A B C A B C 0.38
  4. A B C A B C A B C 0.43

5.21.

  1. A B C 38
  2. A B C 78
  3. A B C 56

5.22. There should be 9 women executives.

5.23. The probability that the students select the same tire is 116 . .

5.26.

  1. Q 1 Q 1 H 1 113 , H 1 H 1 Q 1 14 ,
  2. Q 2 Q 2 H 2 113 , H 2 H 2 Q 2 14 ,
  3. Q 1 Q 2 113 , Q 2 Q 1 Q 1 Q 2 117 .
  4. H 1 H 2 14 , H 2 H 1 H 1 H 2 417 ,
  5. Q 1 Q 1 H 2 113 , H 2 H 2 Q 1 14 ,
  6. Q 2 Q 2 H 1 113 , H 1 H 1 Q 2 14 ,

5.28. Let A denote the event of at least one six. A 1 56 5 0.5981

5.29. Let A denote the event of at least one double six. A 1 3536 10 0.2455

5.31. Let F denote the event that a sum of 4 occurs before a sum of 7. F 13

5.32.

Y k 32243 k 0 80243 k 1 80243 k 2 40243 k 3 10243 k 4 1243 k 5

5.37.

  1. X Y 1112
  2. X 20 Y 20 827

5.42.

  1. R 0.504
  2. R 0.902
  3. R 0.994

5.43. The 5-engine plane would be preferable if p 12 (which one would hope would be the case). The 3-engine plane would be preferable if p 12 . If p 12 , the 3-engine and 5-engine planes are equally reliable.

5.44. Consider cases, depending on whether component 3 is working or failed:

  1. Y X 1 X 2 X 3 X 4 X 5 X 3 X 1 X 2 X 1 X 2 X 4 X 5 X 4 X 5 1 X 3 X 1 X 4 X 2 X 5 X 1 X 2 X 4 X 5
  2. R p 1 p 2 p 3 p 4 p 5 p 3 p 1 p 2 p 1 p 2 p 4 p 5 p 4 p 5 1 p 3 p 1 p 4 p 2 p 5 p 1 p 2 p 4 p 5

5.45. Let L denote the event that the conditions are low stress and W the event that the system works

  1. W 0.9917
  2. L W 0.504

5.48. Let A denote the event that the woman is pregnant and T i the event that test i is positive. A T 1 T 2 T 3 0.834

5.49.

  1. sensitivity 1 1 a 3 , specificity b 3 .
  2. sensitivity 3 a 2 1 a a 3 , specificity b 3 3 b 2 1 b .
  3. sensitivity a 3 , specificity 1 1 b 3 .

5.50. Let C denote the event that the defendant is convicted and G the event that the defendant is guilty.

  1. C 0.51458
  2. G C 0.99996
  3. The independence assumption is ridiculous, of course, since jurors collaborate.

5.51.

  1. 9871024
  2. 271024

5.52. Let C denote the event that the mother is a carrier and let S i denote the event that son i is healthy.

  1. S 1 S 2 58
  2. C S 1 S 2 15
  3. S 3 S 1 S 2 910

5.57. 1112 . No, not really.

6. Convergence

6.23. Let H n be the event that toss n results in heads, and T n the event that toss n results in tails.

  1. n H n 1 , n T n 1 if a 0 1
  2. n H n 0 , n T n 1 if a 1
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