Number X is randomly selected from the set of odd numbers and Y is randomly selected from the set of even numbers of the set (1.2.3.4.5.6.7). Let. Z = (X + Y).
What is P(Z = 5) equal to?

Options

A. `(1)/(2)`
B. `(1)/(3)`
C. `(1)/(4)`
D. `(1)/(6)`

Answer

To solve the problem, we need to find the probability \( P(Z = 5) \) where \( Z = X + Y \), \( X \) is an odd number selected from the set of odd numbers, and \( Y \) is an even number selected from the set of even numbers in the set \( \{1, 2, 3, 4, 5, 6, 7\} \).

### Step-by-Step Solution:

1. **Identify the sets of odd and even numbers**:
- The odd numbers in the set \( \{1, 2, 3, 4, 5, 6, 7\} \) are \( \{1, 3, 5, 7\} \).
- The even numbers in the same set are \( \{2, 4, 6\} \).

2. **Count the number of odd and even numbers**:
- There are 4 odd numbers: \( 1, 3, 5, 7 \).
- There are 3 even numbers: \( 2, 4, 6 \).

3. **Determine the possible pairs \( (X, Y) \) such that \( Z = 5 \)**:
- We need to find pairs \( (X, Y) \) where \( X + Y = 5 \).
- Possible pairs:
- If \( X = 1 \), then \( Y = 4 \) (since \( 1 + 4 = 5 \)).
- If \( X = 3 \), then \( Y = 2 \) (since \( 3 + 2 = 5 \)).
- The pairs that satisfy \( Z = 5 \) are \( (1, 4) \) and \( (3, 2) \).

4. **Calculate the probability of each pair**:
- The probability of selecting \( X = 1 \) (from 4 odd numbers) is \( \frac{1}{4} \).
- The probability of selecting \( Y = 4 \) (from 3 even numbers) is \( \frac{1}{3} \).
- Therefore, the probability of the pair \( (1, 4) \) is:
\[ P(1, 4) = P(X = 1) \times P(Y = 4) = \frac{1}{4} \times \frac{1}{3} = \frac{1}{12} \]

- The probability of selecting \( X = 3 \) (from 4 odd numbers) is \( \frac{1}{4} \).
- The probability of selecting \( Y = 2 \) (from 3 even numbers) is \( \frac{1}{3} \).
- Therefore, the probability of the pair \( (3, 2) \) is:
\[ P(3, 2) = P(X = 3) \times P(Y = 2) = \frac{1}{4} \times \frac{1}{3} = \frac{1}{12} \]

5. **Combine the probabilities of the pairs**:
- Since the pairs are independent, we add the probabilities:
\[ P(Z = 5) = P(1, 4) + P(3, 2) = \frac{1}{12} + \frac{1}{12} = \frac{2}{12} = \frac{1}{6} \]

### Final Answer:
Thus, the probability \( P(Z = 5) \) is \( \frac{1}{6} \).

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