91Ó°ÊÓ

In a family of 4 children, what is the probability that there will be exactly two boys?

If a fair coin is tossed four times, what is the probability of at least two heads?

A baseball player has a \(.250\) batting average (one base hit every four times, on the average). Assuming that the binomial distribution is applicable, if he is at bat four times on a particular day, what is (a) the probability that he will get exactly one hit? (b) the probability that he will get at least one hit?

Given that \(40 \%\) of entering college students do not complete their degree programs, what is the probability that out of 6 randomly selected students, more than half will get their degrees?

The probability that a basketball player makes at least one of six free throws is equal to \(0.999936 .\) Find: (a) the probability function of \(\mathrm{X}\), the number of times he scores; (b) the probability that he makes at least three baskets.

The most common application of the binomial theorem in industrial work is in lot-by-lot acceptance \(\quad\) inspection. If there are a certain number of defectives in the lot, the lot will be rejected as unsatisfactory. It is natural to wish to find the probability that the lot is acceptable even though a certain number of defectives are observed. Let \(\mathrm{p}\) be the fraction of defectives in the lot. Assume that the size of the sample is small compared to the lot size. This will insure that the probability of selecting a defective item remains constant from trial to trial. Now choose a sample of size 18 from a lot where \(10 \%\) of the items are defective. What is the probability of observing 0,1 or 2 defectives in the sample.

An industrial process produces items of which \(1 \%\) are defective. If a random sample of 100 of these are drawn from a large consignment, calculate the probability that the sample contains no defectives.

The probability of hitting a target on a shot is \(2 / 3\). If a person fires 8 shots at a target, Let \(X\) denote the number of times he hits the target, and find: (a) \(\mathrm{P}(\mathrm{X}=3)\) (b) \(\mathrm{P}(1<\mathrm{X} \leq 6)\) (c) \(\mathrm{P}(\mathrm{X}>3)\).

Three electric motors from a factory are tested. A motor is either discarded, returned to the factory or accepted. If the probability of acceptance is \(.7\), the probability of return is \(.2\) and the probability of discard is \(.1\), what is the probability that of three randomly selected motors 1 will be returned 1 will be accepted and 1 will be discarded? What is the probability that 2 motors will be accepted, 1 returned and 0 discarded?