Two person A and B appear in an interview for two vacancies. If the probalities of their sections are `1/4` and `1/6` , respectively, then the probability that none of them is selected is

Options

A. `5/8`
B. `5/12`
C. `1/12`
D. `1/24`

Answer

To find the probability that none of the candidates A and B is selected, we can follow these steps:

### Step 1: Determine the probability of A not being selected.
The probability of A being selected is given as \( P(A) = \frac{1}{4} \). Therefore, the probability of A not being selected is:
\[ P(A') = 1 - P(A) = 1 - \frac{1}{4} = \frac{3}{4} \]

### Step 2: Determine the probability of B not being selected.
The probability of B being selected is given as \( P(B) = \frac{1}{6} \). Therefore, the probability of B not being selected is:
\[ P(B') = 1 - P(B) = 1 - \frac{1}{6} = \frac{5}{6} \]

### Step 3: Calculate the probability that neither A nor B is selected.
Since the selections of A and B are independent events, the probability that neither A nor B is selected is the product of their individual probabilities of not being selected:
\[ P(A' \cap B') = P(A') \times P(B') = \frac{3}{4} \times \frac{5}{6} \]

### Step 4: Perform the multiplication.
Now, we multiply the two fractions:
\[ P(A' \cap B') = \frac{3}{4} \times \frac{5}{6} = \frac{3 \times 5}{4 \times 6} = \frac{15}{24} \]

### Step 5: Simplify the fraction.
To simplify \( \frac{15}{24} \), we find the greatest common divisor (GCD) of 15 and 24, which is 3:
\[ \frac{15 \div 3}{24 \div 3} = \frac{5}{8} \]

### Final Result:
Thus, the probability that none of them is selected is:
\[ \frac{5}{8} \]

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