A bag contains 4 blue and 8 red identical balls. If two balls are picked at random with replacement, what is…

Analyze the question: The bag has 4 blue and 8 red balls, so 12 balls in total. Two balls are picked with replacement, so the two draws are independent. We want the probability that the two balls are different colours.

Concept: "Different colours" can happen in two ways:

• Blue first, then red (BR)
• Red first, then blue (RB)

Add the probabilities of these two cases.

Solution:

P(blue) = 4/12 = 1/3 P(red) = 8/12 = 2/3

P(BR) = 1/3 × 2/3 = 2/9 P(RB) = 2/3 × 1/3 = 2/9

P(different colours) = 2/9 + 2/9 = 4/9

Check: P(same colour) = (1/3)² + (2/3)² = 1/9 + 4/9 = 5/9, and 1 − 5/9 = 4/9. ✓

Tip: In "with replacement" questions, the probabilities do not change between draws. For "different colours" with two colours, remember to count both orders, which means doubling: 2 × (1/3 × 2/3) = 4/9. You can also use the shortcut 1 − P(same colour) to check your answer.