A bag contains 3 red, 3 blue, and 3 green marbles. If 3 marbles are drawn at…

Concept

When all selections are equally likely, the probability of an event is the number of favourable selections divided by the total number of selections.

For an “at least” event, counting its complement is often simpler: P(at least two of one colour) = 1 − P(all three colours are different).

Application

1. Count all unordered selections of 3 marbles from 9: C(9, 3) = 84.

2. For all three colours to be different, choose 1 red, 1 blue, and 1 green marble: C(3, 1) × C(3, 1) × C(3, 1) = 27.

3. Therefore, the number of selections containing at least two marbles of the same colour is 84 − 27 = 57.

4. Hence, the required probability is 57/84 = 19/28.

Cross-check

Direct counting gives the same total: exactly two of one colour can be selected in 3 × C(3, 2) × 6 = 54 ways, and all three of one colour in 3 ways. Thus 54 + 3 = 57, confirming the complement count.

Result

The required probability is 19/28.