Question

It is known that 81% of all new products introduced in grocery stores fail (are taken off the market) within 2 years. If a grocery store chain introduces 69 new products, find the following probabilities. (Round your answers to four decimal places.)

(a) within 2 years 47 or more fail


(b) within 2 years 58 or fewer fail


(c) within 2 years 15 or more succeed


(d) within 2 years fewer than 10 succeed

Answer 1

Solution:

Let X be a random variable which represents the number of new products introduced in a grocery store that fail.

It is known that 81% of all new products introduced in grocery stores fail.

Hence, probability that a new products introduced in grocery stores fail = 81/100 = 0.81

Let us consider finding a new product that fails as success. Hence, we have only two mutually exclusive outcomes, one is success and other is failure.

Hence, probability of success (p) = 0.81

Number of trials (n) = 69

Since, we have only two mutually exclusive outcomes (success and failure) for each trial, number of trials is finite, probability of success remains constant for each trials and outcomes are independent to each other, therefore we can consider that X is binomial distributed random variable.

According to binomial probability law, probability of occurrence of exactly x successes in n trials is given by,

Where, p is probability of success.

a) We have to obtain P(X ≥ 47).

We have, p = 0.81 and n = 69

Using binomial probability law we get,

Using binomial probability law we get,

Using "BINOM.DIST" function of excel we get,

The probability that within 2 years 47 or more new products fail is 0.9966.

b) We have to obtain P(X ≤ 58).

We have, p = 0.81 and n = 69

Using binomial probability law we get,

Using binomial probability law we get,

Using "BINOM.DIST" function of excel we get,

The probability that within 2 years 58 or fewer new products fail is 0.7850.

c) Let Y be a random variable which represents the number of new products introduced in a grocery store that succeed.

It is known that 81% of all new products introduced in grocery stores fail. Hence, 19% of all new products introduced in grocery store succeed.

Hence, probability that a new products introduced in grocery stores succed = 19/100 = 0.19

Let us consider finding a new product that succed as success. Hence, we have only two mutually exclusive outcomes, one is success and other is failure.

Hence, probability of success (p) = 0.19

Number of trials (n) = 69

Since, we have only two mutually exclusive outcomes (success and failure) for each trial, number of trials is finite, probability of success remains constant for each trials and outcomes are independent to each other, therefore we can consider that Y is binomial distributed random variable.

According to binomial probability law, probability of occurrence of exactly y successes in n trials is given by,

Where, p is probability of success.

We have to obtain P(Y ≥ 15).

We have, p = 0.19 and n = 69

Using binomial probability law we get,

Using binomial probability law we get,

Using "BINOM.DIST" function of excel we get,

The probability that within 2 years 15 or more new products succeed is 0.3256.

d) We have to obtain P(Y < 10).

We have, p = 0.19 and n = 69

Using binomial probability law we get,

Using binomial probability law we get,

Using "BINOM.DIST" function of excel we get,

The probability that within 2 years fewer than 10 new products succeed is 0.1318.