A box contains four white marbles and five black marbles. Two marbles are chosen at random without replacement. What is the probability that both marbles are of the same colour?

Introduction / Context:
This question focuses on probabilities with combinations and drawing without replacement. From a box with white and black marbles, two are drawn, and we want the probability that both drawn marbles are of the same colour, either both white or both black.

Given Data / Assumptions:

• White marbles = 4.
• Black marbles = 5.
• Total marbles = 9.
• Two marbles are drawn at random without replacement.
• We are interested in the event that both marbles are white or both are black.

Concept / Approach:
We use combinations to count outcomes. The total number of ways to choose 2 marbles from 9 gives the denominator. The number of ways to choose 2 white marbles plus the number of ways to choose 2 black marbles gives the numerator. The required probability is the ratio of favourable to total outcomes.

Step-by-Step Solution:
Total ways to choose 2 marbles from 9 = 9C2.Compute 9C2 = 9 * 8 / 2 = 36.Favourable case 1: both marbles are white. Ways = 4C2.Compute 4C2 = 4 * 3 / 2 = 6.Favourable case 2: both marbles are black. Ways = 5C2.Compute 5C2 = 5 * 4 / 2 = 10.Total favourable ways = 6 + 10 = 16.Probability both marbles have the same colour = 16 / 36.Simplify 16 / 36 by dividing numerator and denominator by 4 to get 4 / 9.

Verification / Alternative check:
We can also compute the probability sequentially. The probability that the second marble matches the first is: with probability 4/9 the first marble is white and then the second is white with probability 3/8, or with probability 5/9 the first is black and then the second is black with probability 4/8. So the required probability is (4/9)*(3/8) + (5/9)*(4/8) = 12/72 + 20/72 = 32/72 = 4/9, which matches the combination method.

Why Other Options Are Wrong:
The value 2/9 is too small and would correspond to counting only one of the two colour cases or miscomputing combinations. The value 5/9 is too large and ignores the fact that the second draw is conditional on the first. The fraction 1/3 is a rough guess and does not match either method of calculation. Only 4/9 correctly represents the ratio of favourable to total outcomes.

Common Pitfalls:
Some students forget that the draws are without replacement and incorrectly use probabilities such as (4/9)^2 for drawing two whites, which is not correct because the numerator and denominator change on the second draw. Others count only white white pairs and forget black black pairs. Using combinations helps keep track of all valid favourable cases accurately.

Final Answer:
The probability that both marbles are of the same colour is 4/9.