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.
### 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.
