You note that your officer is happy with `60%` of your calls, so, you assign a probability of his being happy on your visit of 0.6 . You have noticed also that if he is happy, he accedes to your request with a probability 0 . 4 , whereas if he is not happy he accedes to the request with a probability of 0.1 . You call one day and he accedes to your request . What is the probability that your office was happy ?

Answer

To solve the problem, we will use Bayes' theorem to find the probability that the officer was happy given that he acceded to your request.

### Step-by-Step Solution:

1. **Define Events:**
- Let \( H \) be the event that the officer is happy.
- Let \( A \) be the event that the officer accedes to your request.

2. **Given Probabilities:**
- Probability that the officer is happy:
\[ P(H) = 0.6 \]
- Probability that the officer is not happy:
\[ P(\bar{H}) = 1 - P(H) = 1 - 0.6 = 0.4 \]
- Probability that the officer accedes to your request given that he is happy:
\[ P(A | H) = 0.4 \]
- Probability that the officer accedes to your request given that he is not happy:
\[ P(A | \bar{H}) = 0.1 \]

3. **Apply Bayes' Theorem:**
We want to find \( P(H | A) \), the probability that the officer is happy given that he accedes to your request. According to Bayes' theorem:
\[ P(H | A) = \frac{P(H) \cdot P(A | H)}{P(A)} \]
where \( P(A) \) can be calculated using the law of total probability:
\[ P(A) = P(A | H) \cdot P(H) + P(A | \bar{H}) \cdot P(\bar{H}) \]

4. **Calculate \( P(A) \):**
\[ P(A) = (0.4 \cdot 0.6) + (0.1 \cdot 0.4) \]
\[ P(A) = 0.24 + 0.04 = 0.28 \]

5. **Substitute Values into Bayes' Theorem:**
Now substitute \( P(A) \) back into the equation for \( P(H | A) \):
\[ P(H | A) = \frac{P(H) \cdot P(A | H)}{P(A)} = \frac{0.6 \cdot 0.4}{0.28} \]
\[ P(H | A) = \frac{0.24}{0.28} \]

6. **Calculate the Final Probability:**
\[ P(H | A) = \frac{0.24}{0.28} \approx 0.857 \]

### Final Answer:
The probability that the officer was happy given that he acceded to your request is approximately \( 0.857 \).