A and B appeared for interview. The probability of their selection is :
`(1)/(3)` and `(1)/(4)` respectively.
Find the probability that :
(i) both selected
(ii) at least one of them selected.

Answer

To solve the problem, we need to calculate two probabilities based on the given probabilities of selection for A and B.

Given:
- Probability of A being selected, \( P(A) = \frac{1}{3} \)
- Probability of B being selected, \( P(B) = \frac{1}{4} \)

### Step 1: Find the probability that both A and B are selected.

To find the probability that both A and B are selected, we can use the formula for the joint probability of independent events:

\[ P(A \text{ and } B) = P(A) \times P(B) \]

Substituting the values:

\[ P(A \text{ and } B) = \frac{1}{3} \times \frac{1}{4} = \frac{1}{12} \]

### Step 2: Find the probability that at least one of them is selected.

To find the probability that at least one of them is selected, we can use the complement rule. The probability that at least one of them is selected is equal to 1 minus the probability that neither of them is selected.

First, we need to find the probability that A is not selected and B is not selected:

\[ P(A') = 1 - P(A) = 1 - \frac{1}{3} = \frac{2}{3} \]
\[ P(B') = 1 - P(B) = 1 - \frac{1}{4} = \frac{3}{4} \]

Now, the probability that neither A nor B is selected is:

\[ P(A' \text{ and } B') = P(A') \times P(B') = \frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2} \]

Now, we can find the probability that at least one of them is selected:

\[ P(\text{at least one of A or B}) = 1 - P(A' \text{ and } B') = 1 - \frac{1}{2} = \frac{1}{2} \]

### Final Answers:
(i) The probability that both A and B are selected is \( \frac{1}{12} \).

(ii) The probability that at least one of them is selected is \( \frac{1}{2} \).