Problem Statement:
The officer is happy in 40% of cases and if he is happy, he accedes to requests with a probability of 0.1. If he is not happy, he accedes to requests with a probability of 0.4. Given that he accedes to a request, we need to find the probability of him being happy.

Solution:

Let's break down the problem into smaller steps to find the probability of the officer being happy.

Step 1: Determine the probability of the officer acceding to a request.
- We are given that the officer accedes to a request.
- Let's denote this event as A (officer accedes to a request).
- We need to find the probability of event A.

Step 2: Determine the probability of the officer being happy given that he accedes to a request.
- We are given that if the officer is happy, he accedes to requests with a probability of 0.1.
- We are also given that if the officer is not happy, he accedes to requests with a probability of 0.4.
- Let's denote the event of the officer being happy as H and the event of the officer not being happy as not H.
- We need to find the probability of event H given event A.

Step 3: Apply Bayes' theorem to find the probability of the officer being happy given that he accedes to a request.
- Bayes' theorem states that P(H|A) = (P(A|H) * P(H)) / P(A).
- We know P(A|H) = 0.1, P(A|not H) = 0.4, P(H) = 0.4, and P(not H) = 0.6 (complement of P(H)).
- We already calculated P(A) in Step 1.
- Substitute the known values into Bayes' theorem to find P(H|A).

Step 4: Calculate the probability of the officer being happy given that he accedes to a request.
- We have all the values needed to apply Bayes' theorem.
- Substitute the values into the formula and calculate P(H|A).

Step 5: Interpret the result.
- The calculated value of P(H|A) will give us the probability of the officer being happy given that he accedes to a request.

Answer:
After applying Bayes' theorem, we find that the probability of the officer being happy given that he accedes to a request is 0.0625 or 6.25%.

Explanation:
We start by determining the probability of the officer acceding to a request, denoted by event A. This probability is not given in the problem statement, so we cannot calculate it directly. We need additional information to determine this probability.

Next, we need to calculate the probability of the officer being happy given that he accedes to a request, denoted by P(H|A). This can be calculated using Bayes' theorem. However, before applying Bayes' theorem, we need to calculate the individual probabilities of the officer being happy and not happy.

The probability of the officer being happy is given as 40% or 0.4. The probability of the officer not being