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Playing Card Frequencies

A standard deck of 52 playing cards consists of 4 suits, with 13 kinds in each suit. In many card games, the kinds are ranked, and are often referred to as the ranks of the cards. In some games, the suits are also ranked.

Pictures of the 52 playing cards
Image from www.jfritz.com/cards

In the picture above, the four rows are the four suits. The clubs are all in the first row, followed by the spades, then the hearts, and last the diamonds. Among the 13 kinds, we find the numbers 2 through 10, and four other kinds. The A stands for ace, the J for jack, the Q for queen, and the K for king. The jack, queen, and king are often referred to as face cards.

In many card games, a player has a number of cards, and this is referred to as his hand. In a few card games, the order in which the cards are received will matter, but more often a player receives all of the cards for his hand at one time. It should be noted that when cards make up a hand, the same card cannot appear twice. In other words, selections of cards to create a hand are done "without replacement" (that is, without returning the first card to the deck and replacing it with a second).

Determining Frequencies of Different Events

The number of ways any particular event can happen will depend upon the number of cards that form a hand. Let us begin with the simplest possible hand, a single card. (Most players would not even consider this a hand, since it has only one card, but mathematicians always include the extreme cases when creating their definitions.)

When a hand consists of multiple cards, the events are much more interesting. Suppose a hand consists of three cards. There are   ${}_{52} C_3 = 22100$ ways to choose such a hand.

Let us consider a single suit, say hearts.

It can be verified that the four results above in fact add to 22100, the total number of ways that something can happen. The fact that we used hearts as the suit was irrelevant, the same frequencies would occur if the suit had been spades (or diamonds, or clubs).

We can do variations on that theme as well.

We can also count various groupings of suits, without identifying specific suits.

Again, we can verify that these three values do add to 22100. Listing all possibilities on a theme, then checking to see that all of the possible combinations have been accounted for, is a very effective way of avoiding errors in your computations.

Now let us consider a single kind, say queens.

These four values also add to 22100.

We can also consider a single kind, without identifying the specific card being sought.

And as before, we can verify that these three values add up to 22100.