If x and y are two positive integers and `x + y = 5`, then what is the probability that x equal 1 ?

Options

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

Answer

To solve the problem, we need to find the probability that \( x = 1 \) given that \( x + y = 5 \) where both \( x \) and \( y \) are positive integers.

### Step-by-Step Solution:

1. **Identify the equation**: We start with the equation \( x + y = 5 \).

2. **Determine the possible values for \( x \) and \( y \)**: Since both \( x \) and \( y \) are positive integers, we can list the possible pairs \((x, y)\) that satisfy the equation:
- If \( x = 1 \), then \( y = 5 - 1 = 4 \) → Pair: (1, 4)
- If \( x = 2 \), then \( y = 5 - 2 = 3 \) → Pair: (2, 3)
- If \( x = 3 \), then \( y = 5 - 3 = 2 \) → Pair: (3, 2)
- If \( x = 4 \), then \( y = 5 - 4 = 1 \) → Pair: (4, 1)

Note: If \( x = 5 \), then \( y = 0 \), which is not a positive integer. Therefore, we stop here.

3. **List the valid pairs**: The valid pairs of \((x, y)\) are:
- (1, 4)
- (2, 3)
- (3, 2)
- (4, 1)

This gives us a total of 4 pairs.

4. **Count the favorable outcomes**: We are interested in the case where \( x = 1 \). From our list, there is only 1 favorable outcome: (1, 4).

5. **Calculate the probability**: The probability \( P(x = 1) \) is given by the formula:
\[ P(x = 1) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{1}{4} \]

Thus, the probability that \( x = 1 \) is \( \frac{1}{4} \).

### Final Answer:
The probability that \( x = 1 \) is \( \frac{1}{4} \).