A and B try to solve the problem independently. The probability that A solves the problem is `(1)/(2)` and that B solves the problem is `(1)/(3)`. Find the probability that :
(a) Both of them solve the problem
(b) The problem is solved.

Answer

To solve the problem, we will break it down into two parts as per the question.

### Given:
- Probability that A solves the problem, \( P(A) = \frac{1}{2} \)
- Probability that B solves the problem, \( P(B) = \frac{1}{3} \)

### (a) Probability that both A and B solve the problem:

Since A and B are solving the problem independently, the probability that both A and B solve the problem can be calculated using the multiplication rule for independent events:

\[ P(A \text{ and } B) = P(A) \times P(B) \]

Substituting the values:

\[ P(A \text{ and } B) = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6} \]

### (b) Probability that the problem is solved:

The problem can be solved in three scenarios:
1. A solves the problem and B does not solve it.
2. B solves the problem and A does not solve it.
3. Both A and B solve the problem.

We can calculate the probability of the problem being solved using the complement rule. First, we will find the probability that neither A nor B solves the problem.

The probability that A does not solve the problem is:

\[ P(A') = 1 - P(A) = 1 - \frac{1}{2} = \frac{1}{2} \]

The probability that B does not solve the problem is:

\[ P(B') = 1 - P(B) = 1 - \frac{1}{3} = \frac{2}{3} \]

Now, the probability that neither A nor B solves the problem is:

\[ P(A' \text{ and } B') = P(A') \times P(B') = \frac{1}{2} \times \frac{2}{3} = \frac{1}{3} \]

Now, the probability that the problem is solved is:

\[ P(\text{Problem is solved}) = 1 - P(A' \text{ and } B') = 1 - \frac{1}{3} = \frac{2}{3} \]

### Final Answers:
(a) The probability that both A and B solve the problem is \( \frac{1}{6} \).

(b) The probability that the problem is solved is \( \frac{2}{3} \).