MATH440, Additional Example on Calculating Probabilities Using Counting Methods

What is the probability that a 5-card poker hand contains at least one card of each suit?

Method 1 (using counting rules—combinations)

There are ways to choose the suit that will contain two cards. Then there are ways to choose two cards (unordered) from that suit. Similarly, there are ways to choose one card from each of the non-two-card suits. Hence, using the basic principle of counting, the probability of at least one card of each suit is .

Note the answer is incorrect. This incorrect answer double counts certain hands—it imposes an ordering on the two same-suit cards.

Method 2 (using inclusion-exclusion)

This method is actually much more complicated than Method 1; I simply wanted to show you an example of the inclusion-exclusion rule.

Let .

Then . [This might not be obvious at first, but it’s an application of DeMorgan’s Law: .]

Therefore, . Using the inclusion-exclusion proposition, . So .