Question 1164818
Since the sample is selected **without replacement** from a finite population, this problem follows a **Hypergeometric Distribution**.

### Given Parameters:

* **Total population size ():** 75 washers
* **Number of successes in population ():** 5 unacceptable washers
* **Number of failures in population ():** 70 acceptable washers
* **Sample size ():** 10 washers

The probability mass function for the hypergeometric distribution is:


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### (a) Probability that none of the unacceptable washers is in the sample

We are looking for . This means we choose 0 from the 5 unacceptable and 10 from the 70 acceptable washers.



Calculating this:


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### (b) Probability that at least one unacceptable washer is in the sample

This is the complement of "none are unacceptable":


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### (c) Probability that exactly one unacceptable washer is in the sample

We are looking for . This means we choose 1 from the 5 unacceptable and 9 from the 70 acceptable washers.



Calculating this:


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### (d) Mean number of unacceptable washers in the sample

For a hypergeometric distribution, the mean is calculated as:


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### R Code

You can use the built-in `dhyper` (for exact probability) and `phyper` (for cumulative probability) functions in R.

```r
# Parameters
N <- 75  # Total population
K <- 5   # Unacceptable in population
n <- 10  # Sample size
m <- K   # Number of 'success' items (unacceptable)
nn <- N - K # Number of 'failure' items (acceptable)

# (a) Probability X = 0
prob_a <- dhyper(0, m, nn, n)
print(paste("Probability none are unacceptable:", round(prob_a, 4)))

# (b) Probability X >= 1
prob_b <- 1 - dhyper(0, m, nn, n)
# Alternatively: phyper(0, m, nn, n, lower.tail = FALSE)
print(paste("Probability at least one is unacceptable:", round(prob_b, 4)))

# (c) Probability X = 1
prob_c <- dhyper(1, m, nn, n)
print(paste("Probability exactly one is unacceptable:", round(prob_c, 4)))

# (d) Mean
mean_val <- n * (m / N)
print(paste("Mean number of unacceptable washers:", round(mean_val, 4)))

```

Would you like me to calculate the **standard deviation** of unacceptable washers for this sample as well?