Virtual Laboratories > Probability Spaces > 1 2 3 4 5 6 7 [8]

8. Notes

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Selected Answers for Section 2

Answer 2.13.

  1. S = {1, 2, 3, 4, 5, 6}2 .
  2. A = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)}
  3. B = {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}
  4. A union B = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}
  5. A intersect B = {(1, 6)}
  6. Ac intersect Bc = (Aunion B)c = {(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)}

Answer 2.15. Denote the denominations by 1 (ace), 2-10, 11 (jack), 12 (queen), 13 (king) and the suits by 0 (clubs), 1 (diamonds), 2 (hearts), 3 (spades).

  1. S = {1, 2, ..., 13} × {0, 1, 2, 3}.
  2. Q = {(12, 0), (12, 1), (12, 2), (12, 3)}
  3. H = {1, 2, ..., 13} × {2}
  4. Q union H = {(y, z) in S: y = 12 or z = 2}
  5. Q intersect H = {(12, 2}}
  6. Q intersect Hc = {(12, 0), (12, 1}, (12, 3)}

Answer 2.17.

  1. S = [-1/2, 1/2]2 .
  2. A = [-1/2 + r, 1/2 - r]2.
  3. Ac = {(x, y) in S: x < -1/2 + r or x > 1/2 - r or y < -1/2 + r or y > 1/2 + r}

Answer 2.19. S = {1, 2, 3, ...}

Answer 2.20.

  1. S = {(1, 4), (2, 3), (3, 2), (4, 1), (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}
  2. A = {(1, 4), (2, 3), (3, 2), (4, 1)}

Answer 2.21. Let 1 denote heads and 0 tails for a coin toss.

  1. S = {(i1, i2, ..., in): n in {1, 2, 3, 4, 5, 6}, ij in {0, 1} j = 1, ..., n}
  2. A = {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}

Answer 2.23. For the coin, let 1 denote heads and 0 tails.

  1. S = {0, 1} × {1, 2, 3, 4, 5, 6}
  2. A = {0, 1} × {4, 5, 6}

Answer 2.25. For gender, let 0 denote female and 1 male.

S = ({18, 19, ...} × {0, 1} × {1, 2, 3})100.

Answer 2.26. For gender, let 0 denote female and 1 male. For species, let 1 denote tredecula, 2 tredecim, and 3 tredecassini.

  1. S = (0, infinity)4 × {0, 1} × {1, 2, 3}
  2. F = {(x1, x2, x3, x4, y, z) in S: y = 0}
  3. S104 where S is given in (a).

Answer 2.27.

  1. S = {0, 1, 2, 3, ...}6 × (0, infinity).
  2. A = {(n1, n2, n3, n4, n5, n6, w) in S: n1 + n2 + n3 + n4 + n5 + n6 > 57}.
  3. S30 where S is given in (a).

Answer 2.28.

  1. S = {0, 1}5.
  2. A = {(x1, x2, x3, x4, x5) in S: x1 + x2 + x3 + x4 + x5 <= 3}

Answer 2.29.

  1. S = (0, infinity)2.
  2. A = (1000, infinity) × (0, infinity).
  3. B = {(x, y) in S: y > x}.
  4. A B = {(x, y) in S: x > 1000 or y > x}
  5. A intersect B = {(x, y) in S: x > 1000 and y > x}
  6. A intersect Bc = {(x, y) in S: x > 1000 and y <= x}

Selected Answers for Section 3

Answer 3.16.

  1. S = {1, 2, 3, 4, 5, 6}2.
  2. Y(x1, x2) = x1 + x2 for (x1, x2) in S.
  3. U(x1, x2) = min{x1, x2} for (x1, x2) in S.
  4. V(x1, x2) = max{x1, x2} for (x1, x2) in S.
  5. {X1 < 3, X2 > 4} = {(1, 5), (2, 5), (1, 6), (2, 6)}
  6. {Y = 7} = {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}
  7. {U = V} = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}

Answer 3.18. Denote the denominations by 1 (ace), 2-10, 11 (jack), 12 (queen), 13 (king) and the suits by 0 (clubs), 1 (diamonds), 2 (hearts), 3 (spades).

  1. S = {1, 2, ..., 13} × {0, 1, 2, 3}.
  2. U(x, y) = x if x < 10, U(x, y) = 10 otherwise.
  3. {U = 10} = {10, 11, 12, 13) × {0, 1, 2, 3}.

Answer 3.20.

  1. S = [-1/2, 1/2]2 .
  2. Z(x, y) = (x2 + y2)1/2 for (x, y) in S.
  3. {X < Y} = {(x, y) in S: x < y}.
  4. {Z < 1/2} = {(x, y) in S: x2 + y2 < 1/4}

Answer 3.22.

  1. S = {0, 1}3.
  2. X(i1, i2, i3) = i1 + i2 + i3 for (i1, i2, i3) in S.
  3. {X > 1} = {110, 101, 011, 111}

Answer 3.23.

  1. S = (0, infinity)2.
  2. {X <1000} = {(x, y) in S: x < 1000}
  3. {X < Y} = {(x, y) in S: x < y}
  4. {X + Y > 2000} = {(x, y) in S: x + y > 2000}

Answer 3.24.

  1. S = {1, 2, 3, 4, 5, 6}3.
  2. W(x1, x2, x3) = #{i: xi = 6} - 1.

Answer 3.27. Let 1 denote heads and 0 tails for a coin toss.

  1. S = {(i1, i2, ..., in): n in {1, 2, 3, 4, 5, 6}, ij in {0, 1} j = 1, ..., n}
  2. N(i1, i2, ..., in) = n for (i1, i2, ..., in) in S.
  3. X(i1, i2, ..., in) = i1 + ··· + in for (i1, i2, ..., in) in S.

Selected Answers for Section 4

Answer 4.20.

  1. S = {1, 2, 3, 4, 5, 6}2.
  2. If the dice are fair, each outcome in S should have the same probability.
  3. P(A) = 1 / 3
  4. P(B) = 5 / 36
  5. P(A intersect B) = 2 / 36.,
  6. P(A union B) = 5 / 12.
  7. P(B intersect Ac) = 1 / 12.

Answer 4.22. Let D = {1, 2, ..., 13} × {0, 1, 2, 3} denote the deck of cards, where the denominations are 1 (ace), 2-10, 11 (jack), 12 (queen), 13 (king) and the suits are 0 (clubs), 1 (diamonds), 2 (hearts), 3 (spades). 

  1. S = {(x1, x2): x1, x2 in D, x1 and x2 distinct} (2652 outcomes).
  2. Since the cards are well shuffled, each outcome in S should have the same probability.
  3. P(H1) = 1 / 4.
  4. P(H1 intersect H2) = 1 / 17.
  5. P(H1c intersect H2) = 13 / 68.
  6. P(H2) = 1 / 4.
  7. P(H1 union H2) = 15 / 34.

Answer 4.24.

  1. S = [-1/2, 1/2]2 .
  2. Since the coin is tossed "randomly," no region of S should be preferred over any other.
  3. P(A) = (1 - 2r)2.
  4. P(Ac) = 1 - (1 - 2r)2.

Answer 4.26.

  1. A occurs but not B. P(A intersect Bc) = 7 / 30.
  2. A or B occurs. P(A union B) = 29 / 60.
  3. One of the events does not occur. P[(A intersect B)c] = 9 / 10.
  4. Neither event occurs. P[(A union B)c] = 31 / 60.
  5. Either A occurs or B does not occur. P(A union Bc) = 17 / 20.

Answer 4.27.

  1. P(A union B union C) = 0.67.
  2. P[(A union B union C)c] = 0.33.
  3. P[(A intersect Bc intersect Cc) union (Ac intersect B intersect Cc) union (Ac intersect Bc intersect C)] = 0.45
  4. P[(A intersect B intersect Cc) union (A intersect Bc intersect C) union (Ac intersect B intersect C)] = 0.21

Answer 4.28.

  1. S = {(1, 4), (2, 3), (3, 2), (4, 1), (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}
  2. Since the dice are fair, each outcome in S should be equally likely.
  3. P(A) = 2 / 5.

Answer 4.29.

  1. S = {0, 1}3.
  2. Since the coins are fair, the outcomes in S should be equally likely.
  3. P(A) = 1 / 2.
  4. P(B) = 3 / 8.
  5. P(A intersect B) = 1 / 4.
  6. P(A union B) = 5 / 8
  7. P(Ac union Bc) = 3 / 4.
  8. P(Ac intersect Bc) = 3 / 8
  9. P(A union Bc) = 7 / 8.

Answer 4.30. Suppose that the balls are numbered from 1 to 12, with balls 1 to 5 red, balls 6 to 9 green, and balls 10 to 12 blue.

  1. S = {{x, y, z}: x, y, z in {1, 2, ..., 12}, x, y, z distinct} (220 outcomes)
  2. P(A) = 3 / 44.
  3. P(B) = 3 / 11.

Answer 4.31. Suppose that the balls are numbered from 1 to 12, with balls 1 to 5 red, balls 6 to 9 green, and balls 10 to 12 blue.

  1. S = {1, 2, ..., 12}3 (1728 outcomes).
  2. P(A) = 1 / 8.
  3. P(B) = 5 / 24.

Answer 4.33.

  1. P(R) = 13 / 30.
  2. P(T) = 19 / 30.
  3. P(W) = 9 / 30.
  4. P(R intersect T) = 9 / 30.
  5. P(T intersect Wc) = 11 / 30.

Answer 4.34.

  1. P(W) = 37 / 104.
  2. P(F) = 59 / 104.
  3. P(T) = 44 / 104.
  4. P(W intersect F) = 34 / 104.
  5. P(W union T union F) = 85 / 104.

Selected Answers for Section 5

Answer 5.5.

  1. P(A | B) = 2 / 5.
  2. P(B | A) = 3 / 10.
  3. P(Ac | B) = 3 / 5.
  4. P(Bc | A) = 7 / 10.
  5. P(Ac | Bc) = 31 / 45.

Answer 5.6.

  1. P(X1 = 3 | Y = 6) = 1 / 5, P(X1 = 3) = 1 / 6, positively correlated.
  2. P(X1 = 3 | Y = 7) = 1 / 6, P(X1 = 3) = 1 / 6, independent.
  3. P(X1 < 3 | Y > 7) = 1 / 15, P(X1 < 3) = 1 / 3, negatively correlated.

Answer 5.8.

  1. P(Q1) = 1 / 13, P(H1) = 1 / 4, P(Q1 | H1) = 1 / 13, P(H1 | Q1) = 1 / 4, independent.
  2. P(Q1) = 1 / 13, P(Q2) = 1 / 13, P(Q1 | Q2) = 3 / 51, P(Q2 | Q1) = 3 / 51, negatively correlated.
  3. P(Q2) = 1 / 13, P(H2) = 1 / 4, P(Q2 | H2) = 1 / 13, P(H2 | Q2) = 1 / 4, independent..
  4. P(Q1) = 1 / 13, P(H2) = 1 / 4, P(Q1 | H2) = 1 / 13, P(H2 | Q1) = 1 / 4, independent.

Answer 5.10. Let Hi denote the event that card i is a heart and Si the event that card i is a spade.

  1. P(H1 intersect H2 intersect H3) = 11 / 850.
  2. P(H1 intersect H2 intersect S3) = 13 / 850.
  3. P(H1 intersect S2 intersect H3) = 13 / 850.

Answer 5.12. 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. P(D intersect S) = 0.036.
  2. P(S | D) = 0.45
  3. S and D are positively correlated.

Answer 5.13.

  1. P(A intersect Bc)| C) = 1 / 4.
  2. P(A union B | C) = 7 / 12.
  3. P(Ac intersect Bc | C) = 5 / 12.

Answer 5.14.

  1. P(A intersect B) = 1 / 4.
  2. P(A union B) = 7 / 12.
  3. P(B union Ac) = 3 / 4.
  4. P(B | A) = 1 / 2.

Answer 5.15. Let R denote the number of reds and W the weight.

P(R >= 10 | W >= 48) = 10 / 23.

Answer 5.16. Let M denote the event that a cicada is male, U the event that the cicada is treducla, and W the body weight.

  1. P(W >= 0.25 | M) = 2 / 45.
  2. P(W >= 0.25 | U) = 7 / 44.

Answer 5.17. The conditional distribution of (X1, X2) given Y = 7 is uniform on {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}.

Answer 5.18.

  1. P(X > 30) = 2 / 3.
  2. P(X > 45 | X > 30) = 1 / 2.
  3. Given X > 30, X is uniformly distributed on (30, 60).

Answer 5.19.

  1. P(Y > 0 | X < Y) = 3 / 4.
  2. Given (X, Y) in [-1/2 + r, 1/2 - r]2, (X, Y) is uniformly distributed on [-1/2 + r, 1/2 - r]2.

Answer 5.23. Let X denote the die score and H the event that all coin tosses result in heads.

  1. P(H) = 21 / 128.
  2. P(X = i | H) = (64 / 63)(1 / 2i) for i = 1, 2, 3, 4, 5, 6.

Answer 5.25. Let U denote the probability of heads for the randomly selected coin, and H the event that the coin lands heads.

  1. P(H) = 41 / 72
  2. P(U = 1 / 2 | H) = 15 / 41, P(U = 1 / 3 | H) = 8 / 41, P(U = 1 | H) = 18 / 41

Answer 5.26. Let X denote the die score and H the even that the coin lands heads.

  1. P(X = i) = 5 / 24 for i = 1, 6; P(X = i) = 7 / 48 for i = 2, 3, 4, 5.
  2. P(H | X = 4) = 3 / 7, P(T | X = 4) = 4 / 7.

Answer 5.28. Let X denote the production line of the selected item, and D the event that the item is defective.

  1. P(D) = 0.037.
  2. P(X = 1 | D) = 0.541, P(X = 2 | D) = 0.405, P(X = 3 | D) = 0.054

Answer 5.29.

  1. 3.75% of the population is colorblind.
  2. 93.3% of colorblind persons are male.

Answer 5.30. Let Ri denote the event that the ball i is red and Gi the event that ball i is green.

  1. P(R1 intersect R2 intersect G3) = 4 / 35.
  2. P(R2) = 3 / 5.
  3. P(R1 | R2) = 2 / 3.

Answer 5.31. Let G denote the event that the ball is green and U1 the event that urn 1 is chosen.

  1. P(G) = 9 / 20.
  2. P(U1 | G) = 2 / 3.

Answer 5.32. Let G1 denote the event that the ball from urn 1 is green, and G2 the event that the ball from urn 2 is green.

  1. P(G2) = 9 / 25.
  2. P(G1 | G2) = 2 / 3.

Selected Answers for Section 6

Answer 6.1.

  1. P(Q1) = P(Q2) = 1 / 13, P(Q2 | Q1) = P(Q1 | Q2) = 1 / 17. Q1, Q2 are negatively correlated.
  2. P(H1) = P(H2) = 1 / 4, P(H2 | H1) = P(H1 | H2) = 4 / 17. H1, H2 are negatively correlated.
  3. P(Q1) = P(Q1 | H1) = 1 / 13, P(H1) = P(H1 | Q1) = 1 / 4. Q1, H1 are independent.
  4. P(Q2) = P(Q2 | H2) = 1 / 13, P(H2) = P(H2 | Q2) = 1 / 4. Q2, H2 are independent.
  5. P(Q1) = P(Q1 | H2) = 1 / 13, P(H2) = P(H2 | Q1) = 1 / 4. Q1, H2 are independent.
  6. P(Q2) = P(Q2 | H1) = 1 / 13, P(H1) = P(H1 | Q2) = 1 / 4. Q2, H1 are independent.

Answer 6.5. There should be 9 women executives.

Answer 6.11. A, B, C are independent if and only if

  1. P(A intersect B) = P(A)P(B).
  2. P(A intersect C) = P(A)P(C).
  3. P(B intersect C) = P(B)P(C).
  4. P(A intersect B intersect C) = P(A)P(B)P(C).

Answer 6.12. A, B, C, D are independent if and only if

  1. P(A intersect B) = P(A)P(B).
  2. P(A intersect C) = P(A)P(C).
  3. P(A intersect D) = P(A)P(D).
  4. P(B intersect C) = P(B)P(C).
  5. P(B intersect D) = P(B)P(D).
  6. P(C intersect D) = P(C)P(D).
  7. P(A intersect B intersect C) = P(A)P(B)P(C).
  8. P(A intersect B intersect D) = P(A)P(B)P(D).
  9. P(A intersect C intersect D) = P(A)P(C)P(D).
  10. P(B intersect C intersect D) = P(B)P(C)P(D).
  11. P(A intersect B intersect C intersect D) = P(A)P(B)P(C)P(D).

Answer 6.13.

  1. P(A union B union C) = 0.93.
  2. P(Ac intersect Bc intersect Cc) = 0.07.
  3. P[(A intersect Bc intersect Cc) union (Ac intersect B intersect Cc) union (Ac intersect Bc intersect C)] = 0.220.
  4. P[(A intersect B intersect Cc) union (A intersect Bc intersect C) union (Ac intersect B intersect C)] = 0.430.

Answer 6.17.

  1. P[(A intersection B) union C] = 3 / 8.
  2. P[A union Bc union C] = 7 / 8.
  3. P[(Ac intersection Bc) union Cc] = 5 / 6.

Answer 6.18. 1/16

Answer 6.21. Let A denote the event of at least one six.

P(A) = 1 - (5 / 6)5 ~ 0.5981.

Answer 6.22. Let A denote the event of at least one double six.

P(A) = 1 - (35 / 36)10 ~ 0.2455

Answer 6.23.

  1. P(X = 0) = 32 / 243
  2. P(X = 1) = 80 / 243
  3. P(X = 2) = 80 / 243
  4. P(X = 3) = 40 / 243
  5. P(X = 4) = 10 / 243
  6. P(X = 5) = 1 / 243

Answer 6.27.

  1. P(X < Y) = 11 / 12.
  2. P(X > 20, Y > 20) = 8 / 27.

Answer 6.32. Let F denote the event that a sum of 4 occurs before a sum of 7.

P(F) = 1 / 3.

Answer 6.37.

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

Answer 6.38.

R = (p1 + p2 - p1 p2)(p4 + p5 - p4 p5)p3 + (p1 p4 + p2 p5 - p1 p2 p4 p5)(1 - p3)

Answer 6.39. Let L denote the event that the conditions are low stress and W the event that the system works

  1. P(W) = 0.9917
  2. P(L | W) = 0.504

Answer 6.42. Let A denote the event that the woman is pregnant and Ti the event that test i is positive.

P(A | T1 intersect T2c intersect T3) = 0.834.

Answer 6.43.

  1. sensitivity 1 - (1 - a)3, specificity b3.
  2. sensitivity 3a2(1 - a) + a3, specificity b3 + 3b2(1 - b).
  3. sensitivity a3, specificity 1 - (1 - b)3.

Answer 6.44. Let C denote the event that the defendant is convicted and G the event that the defendant is guilty.

  1. P(C) = 0.51458
  2. P(G | C) = 0.99996

Answer 6.55. 11 / 12.

Selected Answers for Section 7

Answer 7.25. Let Hn be the event that toss n results in heads, and Tn the event that toss n results in tails.

  1. P(lim supn Hn) = 1, P(lim supn Tn) = 1 if 0 < a 1.
  2. P(lim supn Hn) = 0, P(lim supn Tn) = 1 if a > 0.

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