Box I: cards 1--30.

Question

Box I: cards 1--30. Box II: cards 31--50. A box is selected at random, a card drawn, found non-prime. Find the probability it came from Box I.

Accepted Answer

Non-primes in 1--30: 20 (30 minus 10 primes). Non-primes in 31--50: 15 (20 minus primes 31,37,41,43,47).

$$P(I\mid\text{non-prime}) = \frac{\frac12\times\frac{20}{30}}{\frac12\times\frac{20}{30}+\frac12\times\frac{15}{20}} = \frac{1/3}{1/3+3/8} = \frac{8}{17}$$

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