A student has to appear in two examinations A and B. The probabilities that the student clears A and B are `2/3` and `3/4` respectively. Let p be the probability that he passes both of them, given that he passes at least one of them, then 5.5p = _____

Answer

To solve the problem, we need to find the probability \( p \) that the student passes both examinations A and B, given that he passes at least one of them.

### Step 1: Define the probabilities
Let:
- \( P(A) = \frac{2}{3} \) (Probability of passing exam A)
- \( P(B) = \frac{3}{4} \) (Probability of passing exam B)

### Step 2: Find the probabilities of failing each exam
The probabilities of failing each exam are:
- \( P(A') = 1 - P(A) = 1 - \frac{2}{3} = \frac{1}{3} \) (Probability of failing exam A)
- \( P(B') = 1 - P(B) = 1 - \frac{3}{4} = \frac{1}{4} \) (Probability of failing exam B)

### Step 3: Calculate the probability of passing both exams
Since the events A and B are independent, the probability of passing both exams is:
\[ P(A \cap B) = P(A) \cdot P(B) = \frac{2}{3} \cdot \frac{3}{4} = \frac{6}{12} = \frac{1}{2} \]

### Step 4: Calculate the probability of passing at least one exam
The probability of passing at least one exam can be calculated using the complement of failing both exams:
\[ P(A' \cap B') = P(A') \cdot P(B') = \frac{1}{3} \cdot \frac{1}{4} = \frac{1}{12} \]
Thus, the probability of passing at least one exam is:
\[ P(\text{at least one}) = 1 - P(A' \cap B') = 1 - \frac{1}{12} = \frac{11}{12} \]

### Step 5: Use the conditional probability formula
We need to find \( p \), which is given by:
\[ p = P(A \cap B | \text{at least one}) = \frac{P(A \cap B)}{P(\text{at least one})} \]
Substituting the values we calculated:
\[ p = \frac{P(A \cap B)}{P(\text{at least one})} = \frac{\frac{1}{2}}{\frac{11}{12}} = \frac{1}{2} \cdot \frac{12}{11} = \frac{6}{11} \]

### Step 6: Calculate \( 5.5p \)
Now, we calculate \( 5.5p \):
\[ 5.5p = 5.5 \cdot \frac{6}{11} = \frac{5.5 \cdot 6}{11} = \frac{33}{11} = 3 \]

Thus, the final answer is:
\[ \boxed{3} \]