Question

Suppose that Stephen is the quality control supervisor for a food distribution company. A shipment containing many thousands of apples has just arrived. Unknown to Stephen, 11% of the apples are damaged due to bruising, worms, or other defects. If Stephen samples 10 apples from the shipment, use the binomial distribution to estimate the probability that his sample will contain at least one damaged apple. Give your answer as a decimal precise to at least four decimal places.

Answer 1

0.6882 is the probability that his sample will contain at least one damaged apple.

Explanation:

We are given the following information:

We treat damaged apple as a success.

P(damaged apple) = 11% = 0.11

Then the number of damaged apple follows a binomial distribution, where

`P(X=x) = \binom{n}{x}.p^x.(1-p)^(n-x)`

where n is the total number of observations, x is the number of success, p is the probability of success.

Now, we are given n = 10 and x = 1

We have to evaluate:

`P(x \geq 1) = 1 - P( x = 0) \\= 1 - \binom{10}{0}(0.11)^0(1-0.11)^((10-1))\\= 1 - 0.3118 = 0.6882`

0.6882 is the probability that his sample will contain at least one damaged apple.