A basket contains 9 red balls, 7 white balls, and 4 black balls. Two balls are drawn at random without replacement. What is the probability that exactly one of the two balls drawn is white?

Introduction / Context:
This question deals with drawing balls from a basket without replacement and asks for the probability that exactly one of the two drawn balls is white. It combines counting methods and basic probability rules, and is a common type of question in aptitude tests involving combinations and conditional reasoning.

Given Data / Assumptions:

• Number of red balls = 9.
• Number of white balls = 7.
• Number of black balls = 4.
• Total balls in the basket = 9 + 7 + 4 = 20.
• Two balls are drawn at random without replacement.
• We want the probability that exactly one of the two balls is white.

Concept / Approach:
There are two equivalent approaches. One is to use combinations directly: count the number of ways to choose exactly one white ball and one non white ball, then divide by the total number of ways to choose any two balls. The other is to use ordered probabilities (drawing white then non white, plus non white then white) and sum them. The combination method is simpler and avoids double counting.

Step-by-Step Solution:
Total number of balls = 20. Total number of ways to choose 2 balls from 20 = C(20,2) = 20 * 19 / 2 = 190. We want exactly one white ball in the pair. Number of white balls = 7, number of non white balls (red or black) = 20 - 7 = 13. Number of ways to choose 1 white = C(7,1) = 7. Number of ways to choose 1 non white = C(13,1) = 13. Total favourable pairs with exactly one white = 7 * 13 = 91. Required probability = favourable / total = 91 / 190.

Verification / Alternative check:
Using the ordered probability approach, the event "exactly one white" can occur in two ways: white first then non white, or non white first then white. The probability of white then non white is (7/20) * (13/19). The probability of non white then white is (13/20) * (7/19). Adding these gives (7 * 13 / (20 * 19)) + (13 * 7 / (20 * 19)) = (2 * 7 * 13) / 380 = 182 / 380, which simplifies to 91 / 190. This matches the combination method exactly.

Why Other Options Are Wrong:
18/95 and 18/190: These fractions suggest 36 favourable ordered cases or 18 unordered cases, which do not match the correct count of 91 favourable unordered pairs.
1/2: This would imply that exactly one white occurs in half of all possible pairs, which is not consistent with the given composition.
None of these: This is incorrect because 91/190 is present and is the correct answer.

Common Pitfalls:
Some learners mistakenly compute the probability of at least one white instead of exactly one white. Others forget that the draws are without replacement and use incorrect denominators. It is also easy to double count pairs when working with ordered outcomes and then converting to unordered pairs. The combination based approach neatly avoids such issues.

Final Answer:
Therefore, the probability that exactly one of the two balls drawn is white is 91/190.