Quick answer: Independent events satisfy P(A and B) = P(A)P(B): knowing one occurred does not change the other's probability. Otherwise use P(A)P(B given A).
Worked example
A fair coin is tossed and a fair die is rolled independently. Find P(head and an even number).
- P(head) = 1/2.
- Even die results are 2, 4, 6, giving probability 3/6 = 1/2.
- Multiply independent probabilities: 1/2 × 1/2 = 1/4.
Check the result
There are 12 equally likely combined outcomes; three contain a head and an even result, so 3/12 = 1/4.
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Common mistake to avoid
Mutually exclusive does not mean independent. Drawing without replacement normally changes the second-draw probability.
Try it yourself
1. Find probability of two sixes in two independent fair die rolls. 2. Draw twice with replacement from 3 red and 2 blue balls; find both-red probability.
Answers and reasoning
1. (1/6)² = 1/36. 2. Replacement restores the composition: (3/5)² = 9/25, assuming random draws.
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