49. In rolling two balanced dice, if the sum of the two values is 7, what 1S the probability that one of the values is 1?
50. Robert feels that the probability that he will get an A on the first physics test is 1/2 and the probability that he will get A's on the first and second physics tests is 1/3. If Robert is correct, what is the conditional probability that he will get an A on the second test, given that he gets an A on the first test?
51. Suppose that five cards are to be drawn at random from a standard deck of 52 cards. If all the cards drawn are red, what is the probability that all of them are hearts?
52. A box contains 10 red, 10 white, and 5 blue balls. A ball is chosen at random from the box. If the ball chosen is not blue, what is the probability that it is white?
53. In two tosses of a coin, what is the conditional probability of two heads if the first toss results in heads?
54. Consider the rolling of two balanced dice. (a) What is the conditional probability that at least one lands on 2, given that the dice land on different numbers? (b) What is the conditional probability that at least one lands on an even number, given that the dice land on different numbers?
55. Given that P(A) = 0.5, P(B) = 0.3, and P(AB) = 0.15, verify the following: (a) P(A|B) = P(A), (b) P(A|Bᶜ) = P(A), (c) P(B|A) = P(B), and (d) P(B|Aᶜ) = P(B).
56. If two events A and B are such that P(A) = 0.5, P(B) = 0.3, and P(AB) = 0.1, find the following: (a) P(A|B), (b) P(B|A), (c) P(A|A U B), (d) P(A|AB), and (e) P(AB|A U B).
57. For a certain population of employees the percentages passing and failing a job competency exam, listed according to sex, were as shown in the accompanying table. That is, of all the people taking the exam, 24% were in the male-pass category, 16% were in the male-fail category, and so forth.
An employee is to be selected randomly from this population. Let A be the event that the employee scores a passing grade on the exam and M be the event that a male is selected. (a) Are the events A and M independent? (b) Are the events AC and F independent?
Outcome, Male (M), Female (F), Total
Pass (A), 24, 36, 60
Fail (Aᶜ), 16, 24, 40
Total , 40, 60, 100
58. Show that in two tosses of a fair coin, if A is the event of heads on the first toss and B is the event of tails on the second toss, then A and B are independent.
59. Show that in three throws of a balanced die, if A is the event of a 1 on the first throw, B is the event of a 4 on the second throw, and C is the event of an even number on the third throw, then events A, B, and C are independent.
60. Prove that if A and B are independent events, then A and B are independent.
61. Prove that if A and B are independent events, then Aᶜ and Bᶜ are independent.
62. Let events A and B be independent with P(A) = 0.5 and P(B) = 0.8. Find the probability that neither of the events A or B occurs.
63. Let A and B be events with positive probabilities. Is it possible for these events to be both independent and mutually exclusive? Explain in good detail!
64. Let A and B be events with positive probabilities. Is it possible for these events to be neither independent nor mutually exclusive? Explain in good detail!
65. One-third of the applicants for a certain job are college graduates, and one-fourth of them are college graduates with at least one year of experience in their field. If an applicant is selected at random, what is the probability that if the person is a college graduate he or she also has at least one year of experience?
66. Prove that P(Aᶜ | B) = 1 - P(A|B).
67. Prove that if A and B are independent, then P(A|B) = P(A)
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