In solving any problem, odds against A are 4 to 3 and in favour of Bin solving the same is 7 to 5. The probability that problem will be solved is

Options

A. `(5)/(21)`
B. `(16)/(21)`
C. `(15)/(84)`
D. `(69)/(84)`

Answer

To solve the problem, we need to find the probability that a problem will be solved given the odds against A and in favor of B.

### Step-by-Step Solution:

1. **Understanding Odds Against A**:
- The odds against A are given as 4 to 3. This means:
- Unfavorable cases for A = 4
- Favorable cases for A = 3
- Total cases for A = Unfavorable + Favorable = 4 + 3 = 7.
- Therefore, the probability that A solves the problem (P(A)) is:
\[ P(A) = \frac{\text{Favorable cases for A}}{\text{Total cases for A}} = \frac{3}{7} \]

**Hint**: Remember that odds against A means the first number represents how many times A does not succeed, while the second number represents how many times A does succeed.

2. **Understanding Odds in Favor of B**:
- The odds in favor of B are given as 7 to 5. This means:
- Favorable cases for B = 7
- Unfavorable cases for B = 5
- Total cases for B = Favorable + Unfavorable = 7 + 5 = 12.
- Therefore, the probability that B solves the problem (P(B)) is:
\[ P(B) = \frac{\text{Favorable cases for B}}{\text{Total cases for B}} = \frac{7}{12} \]

**Hint**: Odds in favor of B means the first number represents how many times B succeeds, while the second number represents how many times B fails.

3. **Calculating the Probability of Not Solving**:
- The probability that A does not solve the problem (P(A')) is:
\[ P(A') = \frac{\text{Unfavorable cases for A}}{\text{Total cases for A}} = \frac{4}{7} \]
- The probability that B does not solve the problem (P(B')) is:
\[ P(B') = \frac{\text{Unfavorable cases for B}}{\text{Total cases for B}} = \frac{5}{12} \]

**Hint**: The probability of not solving is simply the complement of the probability of solving.

4. **Finding the Total Probability of Solving the Problem**:
- The total probability that the problem will be solved can be calculated using the formula:
\[ P(\text{solved}) = P(A) + P(B) - P(A) \cdot P(B) \]
- Substituting the values we found:
\[ P(\text{solved}) = \frac{3}{7} + \frac{7}{12} - \left(\frac{3}{7} \cdot \frac{7}{12}\right) \]

5. **Calculating Each Term**:
- First, calculate \( P(A) \cdot P(B) \):
\[ P(A) \cdot P(B) = \frac{3}{7} \cdot \frac{7}{12} = \frac{21}{84} \]
- Now, find a common denominator for \( \frac{3}{7} \) and \( \frac{7}{12} \). The LCM of 7 and 12 is 84.
- Convert \( \frac{3}{7} \) to \( \frac{36}{84} \) (since \( 3 \times 12 = 36 \)).
- Convert \( \frac{7}{12} \) to \( \frac{49}{84} \) (since \( 7 \times 7 = 49 \)).
- Now substitute back into the equation:
\[ P(\text{solved}) = \frac{36}{84} + \frac{49}{84} - \frac{21}{84} \]
- Combine the fractions:
\[ P(\text{solved}) = \frac{36 + 49 - 21}{84} = \frac{64}{84} \]

6. **Simplifying the Result**:
- Simplifying \( \frac{64}{84} \):
\[ P(\text{solved}) = \frac{16}{21} \]

### Final Answer:
The probability that the problem will be solved is \( \frac{16}{21} \).