A bag contains 6 white balls and 4 black balls. Two balls are drawn at random without replacement. What is the probability that the two balls drawn are of the same colour?

Introduction / Context:
This is a basic probability question involving drawing balls from a bag without replacement. The task is to compute the probability that both drawn balls are of the same colour. Such questions check understanding of combinations, total possible outcomes versus favourable outcomes, and the concept of drawing without replacement, which affects the sample space on the second draw.

Given Data / Assumptions:

The bag contains 6 white balls and 4 black balls.

A total of 2 balls are drawn at random without replacement.

All balls of the same colour are indistinguishable for counting purposes.

Each pair of balls drawn is equally likely.

Concept / Approach:
The total number of ways to draw 2 balls from 10 (6 white + 4 black) is given by a combination. The event that both balls are of the same colour can happen in two mutually exclusive ways: both white or both black. We count the combinations for each of these cases separately and add them to get the total favourable outcomes. The probability is then favourable outcomes divided by total outcomes. Using combinations simplifies the counting significantly and avoids ordering issues.

Step-by-Step Solution:
Total number of balls = 6 + 4 = 10.Total number of ways to choose 2 balls from 10 = C(10,2) = 10 * 9 / 2 = 45.Case 1: Both balls are white.Number of ways to choose 2 white balls from 6 = C(6,2) = 6 * 5 / 2 = 15.Case 2: Both balls are black.Number of ways to choose 2 black balls from 4 = C(4,2) = 4 * 3 / 2 = 6.Total favourable outcomes for same colour = 15 + 6 = 21.Therefore, required probability = favourable / total = 21 / 45.Simplify 21/45 by dividing numerator and denominator by 3 to get 7/15.
Verification / Alternative check:
We can also compute using step by step probabilities.Probability both white = (6/10) * (5/9) = 30/90 = 1/3.Probability both black = (4/10) * (3/9) = 12/90 = 2/15.Total probability for same colour = 1/3 + 2/15.Convert 1/3 to 5/15, so total = 5/15 + 2/15 = 7/15, confirming our earlier result.
Why Other Options Are Wrong:
1/2 and 8/15 are larger than the correct probability and occur when counting is done incorrectly, for example by double counting or ignoring some outcomes. 1/9 and 2/5 are smaller than the correct value and reflect mistakes in either the total number of outcomes or the favourable cases. Only 7/15 matches both the combinatorial and step by step calculation.

Common Pitfalls:
One common error is to mix up combinations with permutations and count ordered pairs instead of unordered selections. Another is to treat the second draw as independent of the first despite the words "without replacement"; this changes the denominators and leads to wrong probabilities. Being careful to reflect the changing total number of balls on the second draw is essential for correct results.

Final Answer:
The probability that both balls drawn are of the same colour is 7/15.