Question

In a batch of 8,000 clock radios 4% are defective. A sample of 7 clock radios is randomly selected without replacement from the 8,000 and tested. The entire batch will be rejected if at least one of those tested is defective. What is the probability that the entire batch will be rejected?

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Answer 1

Using the hypergeometric distribution, there is a 0.2486 = 24.86% probability that the entire batch will be rejected.

What is the hypergeometric distribution formula?

The formula is:

$P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}$

$C_{n,x} = \frac{n!}{x!(n-x)!}$

The parameters are:

• x is the number of successes.

• N is the size of the population.

• n is the size of the sample.

• k is the total number of desired outcomes.

For this problem, the values of the parameters are given as follows:

N = 8000, n = 7, k = 0.04 x 8000 = 320

The probability that the entire batch will be rejected is $P(X \geq 1)$, given as follows:

$P(X \geq 1) = 1 - P(X = 0)$

In which:

$P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}$

$P(X = 0) = h(0,8000,7,320) = \frac{C_{320,0}C_{7680,7}}{C_{8000,320}} = 0.7514$

Then:

$P(X \geq 1) = 1 - P(X = 0) = 1 - 0.7514 = 0.2486$

0.2486 = 24.86% probability that the entire batch will be rejected.

More can be learned about the hypergeometric distribution at https://brainly.com/question/24826394

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