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Poker Odds & Probabilities

To be a truly good poker player a fair knowledge of the laws of probability are essential. You should at least know the chances of making up a particular hand. If you are seated at a poker table and have drawn a flush, it would be very helpful to know what the chances are that one of your opponents has put together a superior hand.

No doubt this sounds complicated, and some of the mathematics involved will be beyond the average poker player's immediate ability, having no calculator handy and being pressured by his or her fellow players to either "put up or shut up". This site is designed to help familiarize the beginning player with the various hands and to give some brief notes on the best poker strategy considering the probabilities involved.

A royal flush, strictly speaking, is just a glorified straight flush. A royal flush in spades is considered the highest possible hand. The chances of achieving it are only 1 in the 2,598,960 possible hands. A great book to teach you how to calculate such numbers for yourself is Probability without Tears by Derek Rowntree. This book would help you with the following calculations:

5/52 X 4/51 X 3/50 X 2/49 X 1/48 = 1/2,598,960

The first fraction is the chance of getting any one of 5 particular cards in the deck (5 in 52, since there are 52 cards in the poker deck). The second fraction is your chance of getting any one of 4 particular cards in the deck on your second card, and so on through the fifth fraction or the fifth card. By multiplying them together you find out what the chances are of coming up with any hand of unique cards. The numerator of our fraction in this case represents our royal flush in spades, while the denominator represents all the combinations possible.

If these odds seem long (many lotteries have better odds), take solace in the fact that your chances of getting one of the four possible Royal Flushes dealt to you are four times as great (1 in 649,740). Now, your chances of actually getting a royal flush in a game of poker depends largely on the game you are playing. In draw poker, if you are dealt four of the cards needed your chance of picking up the final card are actually pretty good, 1/47.

The definition of the straight flush includes the royal flush. It basically is made up of five cards all of the same suit and in exact sequence. Rummy players commonly put such runs together in threes and fours after many draws from the pack. However, in poker there are not so many opportunities to make up a hand. Thus the odds of getting such a hand during a poker game are fairly remote. In looking at the royal-flush we found out that the chance of coming up with any one exact hand, say a royal flush in spades, is 1 in 2,598,960.

There are 40 possible straight flushes made up from the standard deck. This is easily illustrated by showing the possibilities of getting a straight flush in any particular suit. Here are the possible combinations:

A K Q J 10
K Q J 10 9
Q J 10 9 8
J 10 9 8 7
10 9 8 7 6

9 8 7 6 5
8 7 6 5 4
7 6 5 4 3
6 5 4 3 2
5 4 3 2 A

Thus all the combinations in one suit comes to ten. If we multiply this by the four suits we can see that there are 40 out of 2,598,960 chances of being dealt a straight flush. Now divide both aspects of our ratio by 40 to get 1/64,974. These are still long odds but better than those of just getting a royal flush. Remember, too, that if you are dealt four of the cards you need for a straight flush, the odds on getting the other card you need are pretty good, 2 in 47.

"Why 2 in 47?" you may well ask. It is because often when you get four cards in a straight flush, all you need to fill it is one card on either end of the straight. For example, you are dealt the 5,6,7,8 of clubs. If you get either the 4 of clubs or the 9 of clubs you have filled your straight flush. The nice thing about going for the straight flush is that even if you fail in making the "flush" portion of your poker game hand you still have a shot at making the straight portion for which you only need a 4 or 9 in any suit. This raises your chances considerably to 8 in 47.

You may have heard the expression, "He was foolish enough to draw to an inside straight". This expression comes from the fact that if you get the outside cards of a straight flush (5,6,8,9 of clubs for example), the chances of getting the 7 are only 1 in 47 half of those in drawing to an "outside" straight flush and only one-eighth of the chance of drawing to a simple straight.