You are a new business owner and ordered 200 promotional smart phone cases to give out to clients. The seller has said that 8 of the cases are defective. You unpack the package and inspect 15 of the cases. 1. What is the probability that 10 are defective? 2. What is the probability that all 15 are defective?

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1. We use the hypergeometric probability formula:

$$
P(X=k)=\frac{\binom{D}{k}\binom{N-D}{n-k}}{\binom{N}{n}}
$$

where:
- $$ N = 200 $$ (total number of cases),
- $$ D = 8 $$ (defective cases),
- $$ n = 15 $$ (cases inspected), and
- $$ k $$ is the number of defective cases in our sample.

For $$ k = 10 $$:

$$
P(X=10)=\frac{\binom{8}{10}\binom{192}{15-10}}{\binom{200}{15}}
$$

Since $$ \binom{8}{10}=0 $$ (it is impossible to choose 10 defective cases from only 8 available), we have

$$
P(X=10)=0.
$$

2. For all 15 cases to be defective ($$ k = 15 $$):

$$
P(X=15)=\frac{\binom{8}{15}\binom{192}{15-15}}{\binom{200}{15}}
$$

Here, $$ \binom{8}{15}=0 $$ (again, it is impossible to choose 15 defective cases from only 8 defective ones), hence

$$
P(X=15)=0.
$$
1. The probability that 10 out of 15 inspected cases are defective is 0. 2. The probability that all 15 inspected cases are defective is 0.

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