SOLUTION: A lot of 75 washers contains 5 in which the variability in thickness around the circumference of the washer is unacceptable. A sample of 10 washers is selected at random, without

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Question 1164818: A lot of 75 washers contains 5 in which the variability in thickness around the circumference
of the washer is unacceptable. A sample of 10 washers is selected at random, without
replacement.
(a) What is the probability that none of the unacceptable washers is in the sample?
(b) What is the probability that at least one unacceptable washer is in the sample?
(c) What is the probability that exactly one unacceptable washer is in the sample?
(d) What is the mean number of unacceptable washers in the sample?

Also give R code
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(2189)   (Show Source): You can put this solution on YOUR website!
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:

---
### (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:

---
### (b) Probability that at least one unacceptable washer is in the sample
This is the complement of "none are unacceptable":

---
### (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:

---
### (d) Mean number of unacceptable washers in the sample
For a hypergeometric distribution, the mean is calculated as:

---
### 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?
Answer by ikleyn(53617)   (Show Source): You can put this solution on YOUR website!
.
A lot of 75 washers contains 5 in which the variability in thickness around the circumference
of the washer is unacceptable. A sample of 10 washers is selected at random, without
replacement.
(a) What is the probability that none of the unacceptable washers is in the sample?
(b) What is the probability that at least one unacceptable washer is in the sample?
(c) What is the probability that exactly one unacceptable washer is in the sample?
(d) What is the mean number of unacceptable washers in the sample?
~~~~~~~~~~~~~~~~~~~~~~~~~~ 


(a)  What is the probability that none of the unacceptable washers is in the sample?


     One way to calculate is to use the formula

         P =  =  .


     The numerator   is the number of favorable    choices from 70 good washers taken 5 at a time;
     the denominator is the number of all possible choices from 75 washers taken 5 at a time.


     This is the ratio of two very big numbers (12-digits or 11-digits).


     They can be calculated in the scientific format, but this way does not give the feeling of numbers.
     So, I prefer another (equivalent) formula

         P =  = 0.4786  (rounded).   ANSWER

     This formula is self-explanatory.


     Part (a) is solved.



(b)  What is the probability that at least one unacceptable washer is in the sample?


     This probability is the complement

         P' = 1 - P = 1 - 0.4786 = 0.5214  (rounded).    ANSWER


     Part (b) is solved.



(c)  What is the probability that exactly one unacceptable washer is in the sample?


     One way to calculate is to use the formula

         P = .


     Again, this is the ratio of two very big numbers (12-digits or 11-digits).


     They can be calculated in the scientific format, but this way does not give the feeling of numbers.
     So, I prefer another (equivalent) formula

     P =  = 0.5178  (rounded).   ANSWER


     Part (c) is solved.



(d)  What is the mean number of unacceptable washers in the sample?

     
     It is   = .

Solved. All questions are answered.


/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/


I submit this my solution as an opposition to the post by @CPhill.

I want you see the difference between a true Math solution,
which teaches on how to solve a typical Math problem by a traditional way and how to makes it effectively
in opposite to the post by @CPhill, which is the run of the undebugged computer code
and demonstrates how the Artificial Intelligence should not communicate with a user.



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