Original Maiths practice · Edexcel IGCSE Maths
Two counters of different colours without replacement
A bag contains three red and two blue counters. Two counters are drawn without replacement. Find the probability that one is red and one is blue, in either order. This original Maiths question requires two mutually exclusive paths.
Step-by-step solution
- First consider red then blue. The first probability is 3/5. Removing a red leaves two blue among four remaining counters, so the second probability is 2/4. The path probability is 3/5 × 2/4 = 3/10.
- Now consider blue then red. The first probability is 2/5. Removing a blue leaves three red among four counters, so the second probability is 3/4. This path also has probability 3/10.
- The two orders cannot happen on the same draw sequence. Add their probabilities: 3/10 + 3/10 = 3/5.
Check the answer
The complement is both the same colour. Both red has probability (3/5)(2/4) = 3/10; both blue has probability (2/5)(1/4) = 1/10. Their total is 2/5, leaving 1 − 2/5 = 3/5 for different colours.
Common errors
Counting only red then blue misses a valid order. Keeping a denominator of five on the second draw assumes replacement, which contradicts the question.
Extend your understanding
With replacement, the answer would be (3/5)(2/5) + (2/5)(3/5) = 12/25. Replacement changes the second-stage probabilities, so the two experiments have different answers.
Skill and difficulty
Skill: Tree diagrams. This is a method-practice question: attempt it after learning the corresponding topic, then use the extension to check deeper understanding. This learning label is not an official examination grade.
Related resources
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