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Chapter 3: Q. 3.11 (page 107)
In each of independent tosses of a coin, the coin lands on heads with probability . How large need be so that the probability of obtaining at least one head is at least ?
Probability of allindependent tails has to be less than,
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Repeat Problem 3.84 when each of the 3 players
selects from his own urn. That is, suppose that there are
3 different urns of 12 balls with 4 white balls in each urn.
A parallel system functions whenever at least one of its components works. Consider a parallel system ofcomponents, and suppose that each component works independently with probability . Find the conditional probability that component 1 works given that the system is functioning.
A deck of cards is shuffled and then divided into two halves of 26 cards each. A card is drawn from one of the halves; it turns out to be an ace. The ace is then placed in the second half-deck. The half is then shuffled, and a card is drawn from it. Compute the probability that this drawn card is an ace. Hint: Condition on whether or not the interchanged card is selected
A total of 500 married working couples were polled about their annual salaries, with the following information resulting:
For instance, in 36 of the couples, the wife earned more and the husband earned less than \( 25,000. If one of the couples is randomly chosen, what is
(a) the probability that the husband earns less than \) 25,000 ?
(b) the conditional probability that the wife earns more than \( 25,000 given that the husband earns more than this amount?
(c) the conditional probability that the wife earns more than \) 25,000 given that the husband earns less than this amount?
The color of a person鈥檚 eyes is determined by a single pair of genes. If they are both blue-eyed genes, then the person will have blue eyes; if they are both brown-eyed genes, then the person will have brown eyes; and if one of them is a blue-eyed gene and the other a brown-eyed gene, then the person will have brown eyes. (Because of the latter fact, we say that the brown-eyed gene is dominant over the blue-eyed one.) A newborn child independently receives one eye gene from each of its parents, and the gene it receives from a parent is equally likely to be either of the two eye genes of that parent. Suppose that Smith and both of his parents have brown eyes, but Smith鈥檚 sister has blue eyes.
(a) What is the probability that Smith possesses a blue eyed gene?
(b) Suppose that Smith鈥檚 wife has blue eyes. What is the probability that their first child will have blue eyes?
(c) If their first child has brown eyes, what is the probability that their next child will also have brown eyes?
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