If A and B are two events such that `P(A)=0.6 and P(B)=0.8,` if the greatest value that `P(A//B)` can have is p, then the value of 8p is______.

Explanation

To solve the problem, we need to find the maximum value of \( P(A|B) \) and then calculate \( 8p \) where \( p \) is this maximum value.

### Step-by-Step Solution:

1. **Understanding Conditional Probability**:
The conditional probability \( P(A|B) \) is defined as:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
where \( P(A \cap B) \) is the probability that both events A and B occur.

2. **Given Values**:
We are given:
- \( P(A) = 0.6 \)
- \( P(B) = 0.8 \)

3. **Finding Maximum Value of \( P(A \cap B) \)**:
The maximum value of \( P(A \cap B) \) can be at most the minimum of \( P(A) \) and \( P(B) \). This is because the intersection cannot exceed the probability of either individual event. Thus:
\[ P(A \cap B) \leq \min(P(A), P(B)) \]
Therefore:
\[ P(A \cap B) \leq \min(0.6, 0.8) = 0.6 \]

4. **Substituting into the Conditional Probability Formula**:
To find the maximum value of \( P(A|B) \), we substitute the maximum value of \( P(A \cap B) \) into the conditional probability formula:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \leq \frac{0.6}{0.8} \]
Simplifying this gives:
\[ P(A|B) \leq \frac{6}{8} = 0.75 \]

5. **Setting \( p \)**:
Thus, the greatest value that \( P(A|B) \) can have is:
\[ p = 0.75 \]

6. **Calculating \( 8p \)**:
Now we need to find \( 8p \):
\[ 8p = 8 \times 0.75 = 6 \]

### Final Answer:
The value of \( 8p \) is \( 6 \).

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