91Ó°ÊÓ

What does the expected value of a binomial distribution with \(n\) trials tell you?

In a binomial experiment, is it possible for the probability of success to change from one trial to the next? Explain.

: Syringes The quality-control inspector of a production plant will reject a batch of syringes if two or more defective syringes are found in a random sample of eight syringes taken from the batch. Suppose the batch contains \(1 \%\) defective syringes. (a) Make a histogram showing the probabilities of \(r=0,1,2,3,4,5,6,7\), and 8 defective syringes in a random sample of eight syringes. (b) Find \(\mu .\) What is the expected number of defective syringes the inspector will find? (c) What is the probability that the batch will be accepted? (d) Find \(\sigma\).

Norb and Gary are entered in a local golf tournament. Both have played the local course many times. Their scores are random variables with the following means and standard deviations. $$ \text { Norb, } x_{1}: \mu_{1}=115 ; \sigma_{1}=12 \quad \text { Gary, } x_{2}: \mu_{2}=100 ; \sigma_{2}=8 $$ In the tournament, Norb and Gary are not playing together, and we will assume their scores vary independently of each other. (a) The difference between their scores is \(W=x_{1}-x_{2} .\) Compute the mean, variance, and standard deviation for the random variable \(W\). (b) The average of their scores is \(W=0.5 x_{1}+0.5 x_{2}\). Compute the mean, variance, and standard deviation for the random variable W (c) The tournament rules have a special handicap system for each player. For Norb, the handicap formula is \(L=0.8 x_{1}-2 .\) Compute the mean, variance, and standard deviation for the random variable \(L .\) (d) For Gary, the handicap formula is \(L=0.95 x_{2}-5 .\) Compute the mean, variance, and standard deviation for the random variable \(L\).

The probability that a single radar station will detect an enemy plane is \(0.65\). (a) Quota Problem How many such stations are required to be \(98 \%\) certain that an enemy plane flying over will be detected by at least one station? (b) If four stations are in use, what is the expected number of stations that will detect an enemy plane?