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Chapter 3: Q. 3.30 (page 111)
Show that for any events and ,
Hint: Compute by conditioning on whether F occurs.
The result is that is weighted average between
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and are involved in a duel. The rules of the duel are that they are to pick up their guns and shoot at each other simultaneously. If one or both are hit, then the duel is over. If both shots miss, then they repeat the process. Suppose that the results of the shots are independent and that each shot of will hit with probability , and each shot of will hit with probability . What is
(a) the probability that is not hit?
(b) the probability that both duelists are hit? (c) the probability that the duel ends after the nth round of shots
(d) the conditional probability that the duel ends after the nth round of shots given that is not hit?
(e) the conditional probability that the duel ends after the nth round of shots given that both duelists are hit?
Two percent of women age who participate in routine screening have breast cancer. Ninety percent of those with breast cancer have positive mammographies. Eight percent of the women who do not have breast cancer will also have positive mammographies. Given that a woman has positive mammography, what is the probability she has breast cancer?
Consider a school community of families, with of them having children, Consider the following two methods for choosing a child:
. Choose one of the families at random and then randomly choose a child from that family.
. Choose one of the children at random.
Show that method is more likely than method to result
in the choice of a firstborn child.
Hint: In solving this problem, you will need to show that
To do so, multiply the sums and show that for all pairs , the coefficient of the term is greater in the expression on the left than in the one on the right.
With probability , the present was hidden by mom; with probability , it was hidden by dad. When mom hides the present, she hides it upstairs percent of the time and downstairs percent of the time. Dad is equally likely to hide it upstairs or downstairs.
(a) What is the probability that the present is upstairs?
(b) Given that it is downstairs, what is the probability it was hidden by dad?
In Example , suppose that the new evidence is subject to different possible interpretations and in fact shows only that it is percent likely that the criminal possesses the characteristic in question. In this case, how likely would it be that the suspect is guilty (assuming, as before, that he has the characteristic)?
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