Chapter 1: Problem 10
Let \(0
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After a hard-fought football game, it was reported that, of the 11 starting players, 8 hurt a hip, 6 hurt an arm, 5 hurt a knee, 3 hurt both a hip and an arm, 2 hurt both a hip and a knee, 1 hurt both an arm and a knee, and no one hurt all three. Comment on the accuracy of the report.
From a well-shuffled deck of ordinary playing cards, four cards are turned over one at a time without replacement. What is the probability that the spades and red cards alternate?
Suppose \(\mathcal{D}\) is a nonempty collection of subsets of \(\mathcal{C}\). Consider the collection of events $$ \mathcal{B}=\cap\\{\mathcal{E}: \mathcal{D} \subset \mathcal{E} \text { and } \mathcal{E} \text { is a } \sigma \text { -field }\\} $$ Note that \(\phi \in \mathcal{B}\) because it is in each \(\sigma\) -field, and, hence, in particular, it is in each \(\sigma\) -field \(\mathcal{E} \supset \mathcal{D}\). Continue in this way to show that \(\mathcal{B}\) is a \(\sigma\) -field.
A chemist wishes to detect an impurity in a certain compound that she is making. There is a test that detects an impurity with probability \(0.90\); however, this test indicates that an impurity is there when it is not about \(5 \%\) of the time. The chemist produces compounds with the impurity about \(20 \%\) of the time. A compound is selected at random from the chemist's output. The test indicates that an impurity is present. What is the conditional probability that the compound actually has the impurity?
A bowl contains 10 chips numbered \(1,2, \ldots, 10\), respectively. Five chips are drawn at random, one at a time, and without replacement. What is the probability that two even-numbered chips are drawn and they occur on even- numbered draws?
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