A man and a woman appears in an interview for two vacancies of same post. If the probabilities of their selectiions are `(1)/(4)` and `(1)/(3)`, respectively. Then, what is the probability of selecting only one of them?

Options

A. `(1)/(12)`
B. `(5)/(12)`
C. `(7)/(12)`
D. `(1)/(24)`

Answer

To solve the problem step by step, we need to calculate the probability of selecting only one of the candidates (the man or the woman).

### Step 1: Define the probabilities
Let:
- \( P_m \) = Probability of the man being selected = \( \frac{1}{4} \)
- \( P_w \) = Probability of the woman being selected = \( \frac{1}{3} \)

### Step 2: Calculate the probabilities of not being selected
- The probability of the man not being selected, \( P_m' \) = \( 1 - P_m = 1 - \frac{1}{4} = \frac{3}{4} \)
- The probability of the woman not being selected, \( P_w' \) = \( 1 - P_w = 1 - \frac{1}{3} = \frac{2}{3} \)

### Step 3: Calculate the probability of selecting only one of them
The probability of selecting only one of them can happen in two scenarios:
1. The man is selected and the woman is not selected.
2. The woman is selected and the man is not selected.

Thus, we can express the total probability of selecting only one of them as:
\[ P(\text{only one selected}) = P_m \cdot P_w' + P_w \cdot P_m' \]

### Step 4: Substitute the values
Substituting the values we calculated:
\[ P(\text{only one selected}) = \left(\frac{1}{4} \cdot \frac{2}{3}\right) + \left(\frac{1}{3} \cdot \frac{3}{4}\right) \]

### Step 5: Calculate each term
Calculating the first term:
\[ \frac{1}{4} \cdot \frac{2}{3} = \frac{2}{12} \]

Calculating the second term:
\[ \frac{1}{3} \cdot \frac{3}{4} = \frac{3}{12} \]

### Step 6: Add the two probabilities
Now, we add the two results:
\[ P(\text{only one selected}) = \frac{2}{12} + \frac{3}{12} = \frac{5}{12} \]

### Final Answer
The probability of selecting only one of them is \( \frac{5}{12} \).

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