A batch of five thousand BluRay discs have been produced His

A batch of five thousand Blu-Ray discs have been produced. Historically, the batch will have 200 defective discs. For quality control purposes, 25 discs are randomly selected from the batch without replacement. Let X = number of defective discs selected out of 25. a) Esplain why X is a binomial random variable. Show all checks (b) Show how to use the binomial formula to find P(selecting exactly 4 defective discs) Use the TI to do the following. Write the event in terms of X and show the calculator command (as done in the class notes) (c Find P(at least two defective disc is selected) (d) Find P(at most 5 defective discs are selected). Show the event and the command.

Solution

(a)

This is a binomial distribution because:

The binomial probability distribution is useful when a total of n independent trials are conducted and we want to find out the probability of r successes, where each success has probability p of occurring. There are several things stated and implied in this brief description. The definition boils down to these four conditions:

(b)

P(X=4) = 25C4 * (0.04)^4 * (0.96)^21 = 0.0137 Answer

(c)

P(X>=2) = 1- P(X<2)

=1- ( 25C0 * (0.04)^0 * (0.96)^25 + 25C1 * (0.04)^1 * (0.96)^24) = 0.264 Answer

(d)

P(X<=5)

= 0.735 + 25C2 * (0.04)^2 * (0.96)^23 + 25C3 * (0.04)^3 * (0.96)^22 + 0.0137+ 25C5 * (0.04)^5 * (0.96)^20

= 0.998 Answer