Three numbers are selected at random without replacement from the set of numbers {1, 2, 3, … n}. The conditional probability that the 3rd number lies between the 1st two. If the 1st number is known to be smaller than the 2nd, is

Options

A. `1/6`
B. `2/3`
C. `1/3`
D. `5/6`

Answer

To solve the problem, we need to find the conditional probability that the 3rd number lies between the 1st and 2nd numbers, given that the 1st number is smaller than the 2nd number.

### Step-by-Step Solution:

1. **Understanding the Selection**:
We are selecting 3 numbers from the set {1, 2, 3, ..., n} without replacement. Let's denote the selected numbers as \( A \), \( B \), and \( C \).

2. **Condition Given**:
We know that \( A < B \). This means that the first number \( A \) is smaller than the second number \( B \).

3. **Possible Arrangements**:
There are 3 numbers selected, and we can arrange them in different ways. The possible arrangements of these numbers, considering the condition \( A < B \), are:
- \( A, B, C \)
- \( A, C, B \)
- \( C, A, B \)

These arrangements ensure that \( A \) is always less than \( B \).

4. **Identifying Favorable Outcomes**:
We need to find the arrangements where the 3rd number \( C \) lies between \( A \) and \( B \).
- In the arrangement \( A, C, B \), \( C \) is between \( A \) and \( B \).
- In the arrangements \( A, B, C \) and \( C, A, B \), \( C \) is not between \( A \) and \( B \).

Therefore, there is only **1 favorable outcome** where \( C \) lies between \( A \) and \( B \).

5. **Total Outcomes**:
From the arrangements we considered, there are a total of **3 possible arrangements** that satisfy the condition \( A < B \).

6. **Calculating the Probability**:
The conditional probability \( P \) that the 3rd number lies between the 1st and 2nd numbers is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{1}{3} \]

### Final Answer:
Thus, the conditional probability that the 3rd number lies between the 1st and 2nd numbers, given that the 1st number is smaller than the 2nd number, is \( \frac{1}{3} \).