91Ó°ÊÓ

In any Ai independent of any other \({\rm{Aj}}\)? Answer using the multiplication property for independent events.

If \({\rm{A}}\)and \({\rm{B}}\) are independent events, show that \({{\rm{A}}^\prime }\) and \({\rm{B}}\)are also independent. (Hint: First establish a relationship between \({\rm{P}}\left( {{{\rm{A}}^{\rm{¢}}}{\rm{ÇB}}} \right){\rm{,P(B)}}\), and \(\left. {{\rm{P(AÇB)}}{\rm{.}}} \right)\)

The proportions of blood phenotypes in the U.S. population are as follows:

Assuming that the phenotypes of two randomly selected individuals are independent of one another, what is the probability that both phenotypes are O? What is the probability that the phenotypes of two randomly selected individual’s match?

In October, \({\rm{1994}}\), a flaw in a certain Pentium chip installed in computers was discovered that could result in a wrong answer when performing a division. The manufacturer initially claimed that the chance of any particular division being incorrect was only \({\rm{1}}\) in \({\rm{9}}\) billion, so that it would take thousands of years before a typical user encountered a mistake. However, statisticians are not typical users; some modern statistical techniques are so computationally intensive that a billion divisions over a short time period is not outside the realm of possibility. Assuming that the \({\rm{1}}\) in \({\rm{9}}\) billion figure is correct and that results of different divisions are independent of one another, what is the probability that at least one error occurs in one billion divisions with this chip?

An aircraft seam requires \({\rm{25}}\) rivets. The seam will have to be reworked if any of these rivets is defective. Suppose rivets are defective independently of one another, each with the same probability.

a. If \({\rm{15\% }}\)of all seams need reworking, what is the probability that a rivet is defective?

b. How small should the probability of a defective rivet be to ensure that only \({\rm{10\% }}\) of all seams need reworking?

A boiler has five identical relief valves. The probability that any particular valve will open on demand is .96. Assuming independent operation of the valves, calculate P(at least one valve opens) and P(at least one valve fails to open).

Two pumps connected in parallel fail independently of one another on any given day. The probability that only the older pump will fail is \({\rm{.10}}\), and the probability that only the newer pump will fail is \({\rm{.05}}\). What is the probability that the pumping system will fail on any given day (which happens if both pumps fail)?

An academic department has just completed voting by secret ballot for a department head. The ballot box contains four slips with votes for candidate Aand three slips with votes for candidate B.Suppose these slips are removed from the box one by one.

a. List all possible outcomes.

b. Suppose a running tally is kept as slips are removed. For what outcomes does Aremain ahead of B throughout the tally?

Consider the system of components connected as in the accompanying picture. Components \({\rm{1}}\) and \({\rm{2}}\) are connected in parallel, so that subsystem works iff either 1 or 2 works; since \({\rm{3}}\)and\({\rm{4}}\) are connected in series, that subsystem works iff both\({\rm{3}}\)and\({\rm{4}}\)work. If components work independently of one another and P(component i works) \({\rm{ = }}{\rm{.9}}\)for \({\rm{i = }}{\rm{.1,2}}\)and \({\rm{ = }}{\rm{.8}}\)for \({\rm{i = 3,4}}\),calculate P(system works).

Refer back to the series-parallel system configuration introduced in, and suppose that there are only two cells rather than three in each parallel subsystem eliminate cells 3 and\({\bf{6}}\), and renumber cells \(4\) and \(5\) as \(3\) and \(4\). The probability that system lifetime exceeds t0 is easily seen to be \(.{\bf{9639}}\). To what value would \(.9\) have to be changed in order to increase the system lifetime reliability from \(.{\bf{9639}}\)to \(.{\bf{99}}\)? (Hint: Let P(Ai ) 5 p, express system reliability in terms of p, and then let x 5 p2 .)