The odds against A solving a certain problem are 3 to 2 and the odds in favour of B solving the same are 2 to 1. The probability that the problem will be solved if they both try, is

Options

A. `(2)/(5)`
B. `(11)/(15)`
C. `(4)/(5)`
D. `(2)/(3)`

Answer

To solve the problem, we need to find the probability that the problem will be solved if both A and B try to solve it.

### Step-by-Step Solution:

1. **Understanding Odds Against A**:
- The odds against A solving the problem are given as 3 to 2. This means that for every 3 times A does not solve the problem, there are 2 times that A does solve it.
- To convert odds against to probability, we can use the formula:
\[ P(A \text{ solves}) = \frac{\text{Odds in favor of A}}{\text{Total Odds}} = \frac{2}{2 + 3} = \frac{2}{5} \]

2. **Calculating Probability of A Not Solving the Problem**:
- The probability that A does not solve the problem is:
\[ P(A \text{ does not solve}) = 1 - P(A \text{ solves}) = 1 - \frac{2}{5} = \frac{3}{5} \]

3. **Understanding Odds in Favor of B**:
- The odds in favor of B solving the problem are given as 2 to 1. This means that for every 2 times B solves the problem, there is 1 time that B does not solve it.
- To convert odds in favor to probability, we can use the formula:
\[ P(B \text{ solves}) = \frac{\text{Odds in favor of B}}{\text{Total Odds}} = \frac{2}{2 + 1} = \frac{2}{3} \]

4. **Calculating Probability of B Not Solving the Problem**:
- The probability that B does not solve the problem is:
\[ P(B \text{ does not solve}) = 1 - P(B \text{ solves}) = 1 - \frac{2}{3} = \frac{1}{3} \]

5. **Calculating the Probability that the Problem is Not Solved by Both**:
- Since A and B are independent in solving the problem, the probability that neither A nor B solves the problem is:
\[ P(A \text{ does not solve} \cap B \text{ does not solve}) = P(A \text{ does not solve}) \times P(B \text{ does not solve}) = \frac{3}{5} \times \frac{1}{3} = \frac{3}{15} = \frac{1}{5} \]

6. **Finding the Probability that the Problem is Solved**:
- The probability that at least one of them solves the problem is:
\[ P(\text{Problem is solved}) = 1 - P(A \text{ does not solve} \cap B \text{ does not solve}) = 1 - \frac{1}{5} = \frac{4}{5} \]

### Final Answer:
The probability that the problem will be solved if both A and B try is \( \frac{4}{5} \).