Exercise 1.35

Question

A random 13-card hand is dealt from a standard deck of cards. What is the probability
that the hand contains at least 3 cards of every suit?

fifthist x · Accepted Answer

We can generate a random hand of $13$ cards with the desired property by 
the following process:

\begin{enumerate}
  \item Pick a suite to sample $4$ cards from

\item Sample $3$ cards for each one of the other suites
\end{enumerate}

There are $4$ suites and $\binom{13}{4}$ ways to sample $4$ cards for any 
of one of them.

By the multiplication rule, there are $\binom{13}{3}^{3}$ ways to sample $3$ 
cards of every one of the remaining $3$ suits.

By the multiplication rule, the total number of possibilities is 
$4 \binom{13}{4} \binom{13}{3}^{3}.$

The unconstrained number of $13$-card hands is $\binom{52}{13}$.

Since each hand is equally likely, by the naive definition of probability, 
the desired likelihood is

$$ \frac{4 \binom{13}{4} \binom{13}{3}^{3}}{\binom{52}{13}} $$

Answers