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Chapter 4: Problem 18
You have three groups of distinctly different items, four in the first group, seven in the second, and three in the third. If you select one item from each group, how many different triplets can you form?
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The two stars of the Miami Heat professional basketball team are very different when it comes to making free throws. ESPN.com reports that Jason Williams makes about \(80 \%\) of his free throws, while Shaquille O'Neal makes only \(53 \%\) of his free throws. \({ }^{4}\) Assume that the free throws are independent, and that each player takes two free throws during a particular game. a. What is the probability that Jason makes both of his free throws? b. What is the probability that Shaq makes exactly one of his two free throws? c. What is the probability that Shaq makes both of his free throws and Jason makes neither of his?
A sample is selected from one of two populations, \(S_{1}\) and \(S_{2}\), with probabilities \(P\left(S_{1}\right)=.7\) and \(P\left(S_{2}\right)=.3 .\) If the sample has been selected from \(S_{1}\), the probability of observing an event \(A\) is \(P\left(A \mid S_{1}\right)=.2 .\) Similarly, if the sample has been selected from \(S_{2}\), the probability of observing \(A\) is \(P\left(A \mid S_{2}\right)=.3 .\) a. If a sample is randomly selected from one of the two populations, what is the probability that event A occurs? b. If the sample is randomly selected and event \(A\) is observed, what is the probability that the sample was selected from population \(S_{1} ?\) From population \(S_{2} ?\)
Identify the following as discrete or continuous random variables: a. Increase in length of life attained by a cancer patient as a result of surgery b. Tensile breaking strength (in pounds per square inch) of 1 -inch-diameter steel cable c. Number of deer killed per year in a state wildlife preserve d. Number of overdue accounts in a department store at a particular time e. Your blood pressure
To reduce the cost of detecting a disease, blood tests are conducted on a pooled sample of blood collected from a group of \(n\) people. If no indication of the disease is present in the pooled blood sample (as is usually the case), none have the disease. If analysis of the pooled blood sample indicates that the disease is present, each individual must submit to a blood test. The individual tests are conducted in sequence. If, among a group of five people, one person has the disease, what is the probability that six blood tests (including the pooled test) are required to detect the single diseased person? If two people have the disease, what is the probability that six tests are required to locate both diseased people?
A random variable \(x\) has this probability distribution: $$\begin{array}{l|rrrrrr}x & 0 & 1 & 2 & 3 & 4 & 5 \\\\\hline p(x) & .1 & .3 & .4 & .1 & ? & .05\end{array}$$ a. Find \(p(4)\). b. Construct a probability histogram to describe \(p(x)\). c. Find \(\mu, \sigma^{2},\) and \(\sigma\). d. Locate the interval \(\mu \pm 2 \sigma\) on the \(x\) -axis of the histogram. What is the probability that \(x\) will fall into this interval? e. If you were to select a very large number of values of \(x\) from the population, would most fall into the interval \(\mu \pm 2 \sigma ?\) Explain.
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