23.
In a survey 7 out of 8 men and 5 out of 9 women said they were satisfied with a product. If 3 women
followed by 1 man are the next customers,
a) what is the probability that they are all satisfied with the product?
b) what is the probability that at least one of them is not satisfied with the product?



Answer :

Certainly! Let's solve this step-by-step.

### Part (a) Probability that they are all satisfied with the product

1. Determine the individual probabilities of satisfaction:
- For men: The probability that a man is satisfied is [tex]\( \frac{7}{8} \)[/tex].
- For women: The probability that a woman is satisfied is [tex]\( \frac{5}{9} \)[/tex].

2. Calculate the combined probability:
- There are 3 women followed by 1 man.
- The events of satisfaction are independent, so we can find the probability of all being satisfied by multiplying their individual probabilities.

3. Mathematical calculation:
[tex]\[ \text{Probability (all satisfied)} = \left(\frac{5}{9}\right)^3 \times \left(\frac{7}{8}\right) \][/tex]

Let's break it down:
- For the 3 women: [tex]\( \left(\frac{5}{9}\right)^3 \)[/tex]
[tex]\[ \left(\frac{5}{9}\right) \times \left(\frac{5}{9}\right) \times \left(\frac{5}{9}\right) = \left(\frac{5 \times 5 \times 5}{9 \times 9 \times 9}\right) = \left(\frac{125}{729}\right) \][/tex]
- For the 1 man: [tex]\( \left(\frac{7}{8}\right) \)[/tex]

4. Combine these probabilities:
[tex]\[ \left(\frac{125}{729}\right) \times \left(\frac{7}{8}\right) = \frac{125 \times 7}{729 \times 8} = \frac{875}{5832} \][/tex]

5. Thus, the probability that they are all satisfied with the product is:
[tex]\[ \boxed{\frac{875}{5832}} \][/tex]

### Part (b) Probability that at least one of them is not satisfied with the product

1. Use the complement rule:
- The probability that at least one is not satisfied is the complement of the probability that all are satisfied.
- Complement rule: [tex]\( P(\text{at least one not satisfied}) = 1 - P(\text{all satisfied}) \)[/tex].

2. We already calculated [tex]\( P(\text{all satisfied}) \)[/tex] in part (a):
[tex]\[ P(\text{all satisfied}) = \frac{875}{5832} \][/tex]

3. Apply the complement rule:
[tex]\[ P(\text{at least one not satisfied}) = 1 - \frac{875}{5832} \][/tex]

4. Simplify the expression:
[tex]\[ P(\text{at least one not satisfied}) = \frac{5832 - 875}{5832} = \frac{4957}{5832} \][/tex]

5. Thus, the probability that at least one of them is not satisfied with the product is:
[tex]\[ \boxed{\frac{4957}{5832}} \][/tex]

These calculations provide the required probabilities for parts (a) and (b) of the question.