An event `X` can take place in conjuction with any one of the mutually exclusive and exhaustive events `A ,Ba n dC` . If `A ,B ,C` are equiprobable and the probability of `X` is 5/12, and the probability of `X` taking place when `A` has happened is 3/8, while it is 1/4 when `B` has taken place, then the probabililty of `X` taking place in conjuction with `C` is `5//8` b. `3//8` c. `5//24` d. none of these

Options

A. `5//8`
B. `3//8`
C. `5//24`
D. None of these

Explanation

To solve the problem step by step, we need to find the probability of event X occurring in conjunction with event C, given the probabilities associated with events A and B.

### Step 1: Understand the Given Information
We know that:
- Events A, B, and C are mutually exclusive and exhaustive.
- The probabilities of A, B, and C are equal (equiprobable).
- The overall probability of event X is \( P(X) = \frac{5}{12} \).
- The probability of X given A has occurred is \( P(X|A) = \frac{3}{8} \).
- The probability of X given B has occurred is \( P(X|B) = \frac{1}{4} \).

### Step 2: Set Up the Probabilities of A, B, and C
Since A, B, and C are equally probable, we can denote their probabilities as:
\[ P(A) = P(B) = P(C) = p \]

Since they are mutually exclusive and exhaustive:
\[ P(A) + P(B) + P(C) = 1 \]
This gives us:
\[ 3p = 1 \]
Thus, we find:
\[ p = \frac{1}{3} \]
So, we have:
\[ P(A) = P(B) = P(C) = \frac{1}{3} \]

### Step 3: Use the Law of Total Probability
The total probability of event X can be expressed using the law of total probability:
\[ P(X) = P(A) \cdot P(X|A) + P(B) \cdot P(X|B) + P(C) \cdot P(X|C) \]

Substituting the known values:
\[ \frac{5}{12} = \frac{1}{3} \cdot \frac{3}{8} + \frac{1}{3} \cdot \frac{1}{4} + \frac{1}{3} \cdot P(X|C) \]

### Step 4: Simplify the Equation
Calculating the first two terms:
1. For \( P(A) \cdot P(X|A) \):
\[ \frac{1}{3} \cdot \frac{3}{8} = \frac{3}{24} = \frac{1}{8} \]
2. For \( P(B) \cdot P(X|B) \):
\[ \frac{1}{3} \cdot \frac{1}{4} = \frac{1}{12} \]

Now, we can rewrite the equation:
\[ \frac{5}{12} = \frac{1}{8} + \frac{1}{12} + \frac{1}{3} \cdot P(X|C) \]

### Step 5: Find a Common Denominator
The common denominator for 8, 12, and 3 is 24. Rewriting the fractions:
- \( \frac{1}{8} = \frac{3}{24} \)
- \( \frac{1}{12} = \frac{2}{24} \)
- \( \frac{5}{12} = \frac{10}{24} \)

Substituting these values:
\[ \frac{10}{24} = \frac{3}{24} + \frac{2}{24} + \frac{1}{3} \cdot P(X|C) \]

### Step 6: Combine the Terms
Combining the terms on the right:
\[ \frac{10}{24} = \frac{5}{24} + \frac{1}{3} \cdot P(X|C) \]

### Step 7: Isolate \( P(X|C) \)
Subtract \( \frac{5}{24} \) from both sides:
\[ \frac{10}{24} - \frac{5}{24} = \frac{1}{3} \cdot P(X|C) \]
\[ \frac{5}{24} = \frac{1}{3} \cdot P(X|C) \]

### Step 8: Solve for \( P(X|C) \)
To isolate \( P(X|C) \), multiply both sides by 3:
\[ P(X|C) = 3 \cdot \frac{5}{24} = \frac{15}{24} = \frac{5}{8} \]

### Conclusion
Thus, the probability of event X taking place in conjunction with event C is:
\[ P(X|C) = \frac{5}{8} \]

### Final Answer
The correct option is \( \frac{5}{8} \).