- #1

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- TL;DR Summary
- compute a probability

I have a deck of 60 cards. I draw 7 cards. Among the 60 cards are 21 cards in category "A", 4 cards in category "B" and 8 cards in category "C". The categories are mutually exclusive. I want to the probability that my 7 cards contain at least 2 from category A, 1 from category B and 1 from category C. I count as follows:

$$

\frac{\binom{21}{2}\binom{4}{1}\binom{8}{1}\binom{56}{3}}{\binom{60}{7}}

$$

This is 0.482328 but computer simulations (as well as a sample of 100 hand drawn manually) show that this is off by a factor of 3. The real answer is close to 16%.

Why is my way of counting no good? Where's that factor of 3 coming from ?

$$

\frac{\binom{21}{2}\binom{4}{1}\binom{8}{1}\binom{56}{3}}{\binom{60}{7}}

$$

This is 0.482328 but computer simulations (as well as a sample of 100 hand drawn manually) show that this is off by a factor of 3. The real answer is close to 16%.

Why is my way of counting no good? Where's that factor of 3 coming from ?