statistic question

Question

A brand name has a 50% recognition rate. Assume the owner of the brand wants to verify that rate by beginning with a small sample of 8 randomly selected consumers.

a.) What is the probability that exactly 7of the selected consumers recognize the brand​ name?

The probability that exactly 7of the 8 consumers recognize the brand name is ____

(Round to three decimal places as​ needed.)

b.)  What is the probability that all of the selected consumers recognize the brand​ name?

The probability that all of the selected consumers recognize the brand name is _____

(Round to three decimal places as​ needed.)

Answer 1

Assuming that the number of consumers from which this sample is drawn is quite large, we can model each sample choice as being independent. In general, the probability of exactly h successes out of k trials for a random variable with probability P of success is found by first calculating the probability of one possible permutation, then multiplying by the number of possible permutations for that number of successes.

a.) In this case, our example permutation can be the one in which all trials succeed except the last one:

(.5)⁷(.5)¹ = 1/256 or roughly .00391.

The number of possible permutations is given by

k!/[h!(k-h)!] = 8!/[7!(8-7)!] = 8.

(An easier way to think about this particular example would be to simply observe that there are 8 possible slots for the single "failure".) Multiplying together our two results, we have

8/256 = 1/32 = .03125.

This is the probability of observing any one of the 8 possible ways to get 7 out of 8 successes.

b.) This calculation is significantly simpler, since there's only one way to observe 8 out of 8 successes. Our calculation is simply

(.5)⁸ = 1/256 = roughly .00391.