- #1
- 4,807
- 32
- 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 ?