Three numbers are chosen at random without replacement from {1, 2, 3, ...... 8}. The probability that their minimum is 3, given that their maximum is 6, is (1) `3/8` (2) `1/5` (3) `1/4` (4) `2/5`

Options

A. `3/8`
B. `1/5`
C. `1/4`
D. `2/5`

Explanation

To solve the problem, we need to find the probability that the minimum of three randomly chosen numbers from the set {1, 2, 3, ..., 8} is 3, given that their maximum is 6.

### Step 1: Define the Events
Let:
- Event A: The minimum of the three chosen numbers is 3.
- Event B: The maximum of the three chosen numbers is 6.

We need to find \( P(A | B) \), which is the probability of event A occurring given that event B has occurred.

### Step 2: Use the Conditional Probability Formula
The formula for conditional probability is:
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]
Where \( P(A \cap B) \) is the probability that both events A and B occur.

### Step 3: Determine \( P(A \cap B) \)
For \( P(A \cap B) \), we need to find the number of ways to choose three numbers such that the minimum is 3 and the maximum is 6.

The numbers we can choose from are {3, 4, 5, 6}. Since the minimum must be 3 and the maximum must be 6, the third number must be either 4 or 5.

- If we choose 3 as the minimum and 6 as the maximum, the possible combinations for the third number are:
- Choose 4: The set is {3, 4, 6}
- Choose 5: The set is {3, 5, 6}

Thus, there are 2 combinations that satisfy both conditions:
1. {3, 4, 6}
2. {3, 5, 6}

So, \( n(A \cap B) = 2 \).

### Step 4: Determine \( P(B) \)
Next, we need to find \( P(B) \), which is the probability that the maximum of the three chosen numbers is 6.

If the maximum is 6, the numbers can be chosen from the set {1, 2, 3, 4, 5, 6}. We need to choose 2 more numbers from the set {1, 2, 3, 4, 5} (since 6 is already chosen as the maximum).

The number of ways to choose 2 numbers from the 5 available (1, 2, 3, 4, 5) is given by:
\[ n(B) = \binom{5}{2} = 10 \]

### Step 5: Calculate \( P(A | B) \)
Now we can substitute the values into the conditional probability formula:
\[ P(A | B) = \frac{n(A \cap B)}{n(B)} = \frac{2}{10} = \frac{1}{5} \]

### Conclusion
Thus, the probability that the minimum is 3 given that the maximum is 6 is:
\[ \boxed{\frac{1}{5}} \]