1.3.6 problem 35

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

1.

Pick a suite to sample $4$ cards from

2.

Sample $3$ cards for each one of the other suites

There are $4$ suites and $\left(\frac{13}{4}\right)$ ways to sample $4$ cards for any of one of them.

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

By the multiplication rule, the total number of possibilities is $4\left(\frac{13}{4}\right){\left(\frac{13}{3}\right)}^3.$

The unconstrained number of $13$-card hands is $\left(\frac{52}{13}\right)$.

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