A factory received a shipment of 20 bolts, and the vendor who sold the items knows there are 7 bolts in the shipment that are defective. Before the receiving foreman accepts the delivery, he samples the shipment, and if too many of the bolts in the sample are defective, he will refuse the shipment.

For each of the following, give your responses as reduced fractions. If a sample of 7 bolts is selected, find the probability that none in the sample are defective.


We are given a total of 20 bolts in the shipment, of which 7 are defective. A sample of 7 bolts is selected, and we need to find the probability that none of the bolts in the sample are defective.

This problem involves hypergeometric probability, where the probability of selecting a certain number of defective bolts from a sample without replacement can be calculated using the hypergeometric formula.

The probability of selecting **k** defective bolts in a sample of size **n** from a population of size **N** with **D** defective bolts is given by the hypergeometric distribution formula:

\[
P(\text{exactly } k \text{ defective bolts}) = \frac{\binom{D}{k} \binom{N-D}{n-k}}{\binom{N}{n}}
\]

### In our case:
- \( N = 20 \) (total number of bolts)
- \( D = 7 \) (number of defective bolts)
- \( n = 7 \) (sample size)
- \( k = 0 \) (number of defective bolts we want in the sample)

### Step 1: Calculate the combinations
- \( \binom{7}{0} = 1 \) (ways to choose 0 defective bolts from 7 defective ones)
- \( \binom{13}{7} \) (ways to choose 7 non-defective bolts from 13 non-defective ones)
- \( \binom{20}{7} \) (total ways to choose 7 bolts from 20 bolts)

Now let's compute the necessary values and the probability. I'll do this computation for you.The probability that none of the bolts in the sample are defective is approximately \( 0.0221 \), or as a reduced fraction, \( \frac{53}{2394} \).

Would you like further details or have any questions?

Here are five related questions to expand on this topic:
1. What is the probability that exactly one bolt in the sample is defective?
2. What is the probability that at least one bolt in the sample is defective?
3. How does the hypergeometric distribution differ from the binomial distribution in this context?
4. What happens to the probability if the sample size is increased to 10?
5. What are the expected number of defective bolts in a sample of 7?

**Tip**: When sampling without replacement, consider using the hypergeometric distribution for accurate probability calculations!

A factory received a shipment of 20 bolts, and the vendor who sold the items knows there are 7 bolts in the shipment that are defective. Before the receiving foreman accepts the delivery, he samples the shipment, and if too many of the bolts in the sample are defective, he will refuse the shipment. If a sample of 7 bolts is selected, find the probability that none in the sample are defective.

Calculate the probability of selecting no defective bolts from a sample of 7 in a shipment of 20 bolts, using the hypergeometric probability distribution. This problem involves combinatorics and probability concepts suitable for Grades 10-12.

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