Remedial Math

Being the stud or studette poker player that you are, you probably don’t need a refresher course in basic Texas hold’em math, but since you might accidentally leave this magazine where a newbie may find it, let’s do that earnest striver a favor and review a few fundamental numbers. Accompanying this column you’ll find some charts that even experienced players will find useful to refer to from time to time. Cut ‘em out, stick ‘em in your wallet and review ‘em between hands. With a little bit of study, you won’t need them at all any more, and then you can leave them where the newbies will find them.

In hold’em, as we know, we often need to know the chances of certain cards presenting themselves to us on the flop, turn, or the river. For instance, to figure out if you’re getting the right price to call a bet, you may need to know the probability of catching a key flush card on the river, or flopping that third 8 to go with your pair in hand. In order to compute the chances of finding the card you need on the board, you will need to know two things. The first is how many outs you have, where outs are the number of outstanding cards in the deck that can complete your hand. If you’re chasing a club flush on the river, for example, there are nine cards that can help you. (Thirteen clubs in the deck minus four clubs in your hand or on the board equals nine clubs remaining.)

The second thing you need to know is the number of unseen cards left in the deck. Since two cards are in your hand, and four are on board on the turn, there are 46 cards remaining. (52-2=50 followed by 50-4=46). Remember that even though there are cards in other players’ hands, you don’t know what they are, so even though they are in play, they count as unseen. Now computing the odds becomes a simple problem of long division. (I wonder what short division is; no one ever says.) Nine cards that can help, divided by 46 cards in total, equals 19.6 percent. (9÷46=19.56 percent). You thus have a 19.6 percent chance of making your flush with one card to come. Sometimes, of course, you have to adjust those numbers to take into consideration the cards that may already be in your opponents’ hands. If, for instance, you’re chasing a flush and think that someone else is too, then you have to subtract those two supposed clubs from your outs and from the total cards remaining in the deck. You would calculate that thus: 7÷44=15.22 percent).

It’s better, of course, if you don’t have to do the calculation, if you already know the math of common situations inside out. So here are some numbers you can use right away:

Flush: If you hold four to a flush, the chances of completing are 19.1 percent on the turn and 19.6 percent on the river. Taken together, there’s a 35 percent chance that you’ll complete your flush if you get to see both the turn and river cards. Remember not to assume that you’ll get to see both cards for free, for savvy, strong foes will make you pay to draw on the turn.

Open-Ended Straight: Your chance of catching your card on the turn is 17.0 percent. On the river, it’s 17.4 percent. On the turn or the river combined, it’s 31.5 percent. So you’ll complete your straight draw about one time in three. Again, remember to calculate the percentages street by street. You’re only guaranteed to see the river for free if you or your foes are all-in on the flop.

Gutshot Straight: You’ll catch one of your four outs on the turn 8.5 percent of the time, and on the river 8.7 percent. Filling a gutshot on the turn and river combined happens 16.5 percent of the time.

Sets: You’ll turn your pair into three of a kind 4.3 percent of the time on the turn and 4.3 percent of the time on the river. Note: these numbers are rounded off; you’re functionally as likely to catch trips on the turn as on the river because you’re basically pretty dang unlikely to catch trips at all. Keep the length of this longshot in mind. Catching trips on the turn or river is a combined 8.4 percent shot. The good news is that if you start with a pocket pair, you will already have made a set on the flop 11.8 percent of the time.

In a minute here I’m going to hit you with some charts to finish out this column, but before we get to that, I want to give you a handy shorthand for calculating your odds on the fly. This system works because most of the time you’re counting your outs against a deck that contains just slightly less than half a hundred cards, which allows you to do the following cheaty math: Just multiply your number of outs times the number of cards to come, and then times 2 percent. If you have nine flush outs on the river, that’s 9 × 1 × 2 percent, or 18 percent. The true outs are 19.6 percent but, “close enough for jazz,” as the saying goes. To take another example, if you’re drawing to an open-ended straight and can count on seeing both the turn and river cards, you’re looking at eight outs times two cards to come times two percent (8 × 2 × 2 percent), or 32 percent. Since the exact probability is 31.5 percent, you can see that you’re not too far off. With this handy rule of thumb working for you, you can always be reasonably clear on where you stand on the likelihood of completing your hand.

Okay, here come the charts. Save and store in a cool, dry place.

Poker Hand Odds
Probability of Catching an Out

John Vorhaus is author of the Killer Poker series and co-author of Decide to Play Great Poker, plus many mystery novels including World Series of Murder, available exclusively on Kindle. He tweets for no apparent reason @TrueFactBarFact and secretly controls the world from johnvorhaus.com.