Two brothers X and Y appear independently for a selection test. The probability that X is selected is 1/7 and the probability that Y is selected is 2/9. What is the probability that both brothers are selected?

Introduction / Context:
This is a straightforward probability question involving independent events. Two brothers, X and Y, have given a test and each has a known probability of being selected. We are asked for the probability that both are selected. When two events are independent, the probability that both occur is the product of their individual probabilities.

Given Data / Assumptions:

Probability that X is selected = 1/7.

Probability that Y is selected = 2/9.

The selection outcomes for X and Y are assumed to be independent.

We want the probability that X is selected and Y is selected.

Concept / Approach:
For independent events A and B, the probability that both occur is P(A and B) = P(A) * P(B). Here, event A is "X is selected" and event B is "Y is selected". Because the outcome for one brother does not affect the outcome for the other, we can multiply the two probabilities directly to obtain the required result.

Step-by-Step Solution:
Step 1: Let A be the event that X is selected, so P(A) = 1/7.Step 2: Let B be the event that Y is selected, so P(B) = 2/9.Step 3: Since the events are independent, P(A and B) = P(A) * P(B).Step 4: Compute P(A and B) = (1/7) * (2/9) = 2 / 63.Step 5: Therefore, the probability that both brothers are selected is 2/63.

Verification / Alternative check:
We can also think in terms of a simple grid of outcomes: X may be selected or not selected, and Y may be selected or not selected. The probability for any particular pair of outcomes is the product of the corresponding individual probabilities. The desired outcome is (selected, selected), which has probability 1/7 * 2/9 = 2/63, matching our earlier calculation.

Why Other Options Are Wrong:
1/63 would correspond to incorrectly multiplying 1/7 and 1/9 or misreading the selection probability of Y. The fraction 1/14 is larger than 2/63 and might come from adding probabilities instead of multiplying. The value 1/9 ignores the selection probability of X, and 4/63 would require either both probabilities to be larger or incorrect multiplication. Only 2/63 correctly reflects the product of 1/7 and 2/9.

Common Pitfalls:
A typical mistake is to add probabilities (1/7 + 2/9) instead of multiplying them, which would give the probability that at least one is selected only if we also subtracted the overlap. Another mistake is to treat the events as mutually exclusive, which they are not, because both brothers can be selected simultaneously. Always check whether events are independent or mutually exclusive before choosing the appropriate rule.

Final Answer:
The probability that both X and Y are selected is 2/63.