Question

A bridge hand is made up of 13 cards from a deck of 52. Find the probability that a hand chosen at random contains at least 3 nines.

Answer 1

Let X = number of nines in a 13 bridge hand
We want P( X >= 3) .
The complement approach would be easier.
P( X>= 3 ) = 1 - P( X <3 )
= 1 - P ( X <= 2 )
= 1- ( P(X=0 + P(X=1) + P(X=2) )
$$ \large
\rm {
P(X \leq 2 ) = 1 - \left ( \frac{ \binom {48}{13} \cdot \binom{4}{0}}{\binom {52}{13}} +
\frac{ \binom {48}{12} \cdot \binom{4}{1}}{\binom {52}{13}} +
\frac{ \binom {48}{11} \cdot \binom{4}{2}}{\binom {52}{13}}
\right )



}
$$

Answer 2

Thank you so much Perl!!!! This is very helpful!!!!!! I have become your fan :)

Answer 3

also i might clarify a bit on how i got the fractions, and i am using binomial notation
n choose r \( \binom n r \)

$$ \Large \rm {

\frac{ \binom {not ~nines }{\#} \cdot \binom{nines}{\#}}{\binom {52}{13}}


}
$$

Answer 4

there are 48 'not nines' , and 4 nines