Question

of ten twenty-dollar bills, 2 are counterfeit. six bills are chosen at random. what is the probability that both counterfeit bills are chosen?

Answer 1

\[\frac{1}{3}\] there are two ways of doing this

Answer 2

the number of ways to choose 6 from 10 is
\[\dbinom{10}{6}=210\]
and there is only one way to choose both counterfeit bills and
\[\dbinom{8}{4}=70\] ways to choose the other 4 so the answer is
\[\frac{70}{210}=\frac{1}{3}\]

Answer 3

second way is to say that the probability first is counterfeit is
\[\frac{2}{10}\] and then the probability that the second is counterfeit given the first one is is
\[\frac{1}{9}\] and then the other bills are all good. but there are
\[\dbinom{6}{2}=15\] ways to arrange the counterfeit bills in the set of 6 chosen, so the probability is
\[15\times \frac{2}{10}\times \frac{1}{9}=\frac{1}{3}\]