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Chapter 3: 3.88 (page 106)
In Example 5e, what is the conditional probability that the ith coin was selected given that the first n trials all result in heads?
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(a) An urn containswhite and black balls. The balls are withdrawn one at a time until only those of the same color are left. Show that with probability , they are all white. Hint: Imagine that the experiment continues until all the balls are removed, and consider the last ball withdrawn.
(b) A pond containsdistinct species of fish, which we will call the Red, Blue, and Greenfish. There are Red, Blue, and Greenfish. Suppose that the fish are removed from the pond in random order. (That is, each selection is equally likely to be any of the remaining fish.) What is the probability that the Redfish are the first species to become extinct in the pond?
Hint: Write , and compute the probabilities on the right by first conditioning on the last species to be removed.
Suppose that each child born to a couple is equally likely to be a boy or a girl, independently of the sex distribution of the other children in the family. For a couple having children, compute the probabilities of the following events:
(a) All children are of the same sex.
(b) The eldest are boys and the others girls.
(c) Exactly are boys.
(d) The oldest are girls.
(e) There is at least girl.
A red die, a blue die, and a yellow die (all six-sided) are rolled. We are interested in the probability that the number appearing on the blue die is less than that appearing on the yellow die, which is less than that appearing on the red die. That is, with B, Y, and R denoting, respectively, the number appearing on the blue, yellow, and red die, we are interested in .
(a) What is the probability that no two of the dice land on the same number?
(b) Given that no two of the dice land on the same number, what is the conditional probability that ?
(c) What is ?
Three prisoners are informed by their jailer that one of them has been chosen at random to be executed and the other two are to be freed. Prisoner A asks the jailer to tell him privately which of his fellow prisoners will be set free, claiming that there would be no harm in divulging this information because he already knows that at least one of the two will go free. The jailer refuses to answer the question, pointing out that if A knew which of his fellow prisoners were to be set free, then his own probability of being executed would rise from 1 3 to 1 2 because he would then be one of two prisoners. What do you think of the jailer鈥檚 reasoning?
Let , and be events relating to the experiment of rolling a pair of dice.
(a) If localid="1647938016434" and localid="1647938126689" either prove that localid="1647938033174" or give a counterexample by defining events and for which that relationship is not true.
(b) If localid="1647938162035" and either prove that or give a counterexample by defining events and for which that relationship is not true. Hint: Let be the event that the sum of a pair of dice is ; let be the event that the first die lands on ; let be the event that the second die lands on .
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