Simple Arithmetic for Hold'em Players - Part 2 / By: Lou Krieger
/ Part 1 - Part 2 |
This is the third and final lesson dealing with the mathematics of hold'em. In
the first two articles you learned how to calculate the number of possible
two-card combinations before the flop, as well as the odds against being dealt a
pair of aces - or any other pair, for that matter. Calculating the number of
possible flops, and figuring the chances of flopping a set or better when you
have a pair in your hand should no longer be a mystery.
Changing percentages into odds, and odds to percentages, ought to be a slam-dunk
- and so should problems like this-to- Suppose you're holding a pair of queens
and know that I'll raise only with aces, kings, or A-K, who's the favorite if I
do raise?
With this foundation in hold'em arithmetic, we'll get down to some of the more
important calculations. Flopping a four-flush or a four to a straight happens
regularly when you play hold'em. Knowing the odds against making either of these
hands, when compared with the money odds offered by the pot, provides all the
information needed to determine whether folding or drawing gives you the best of
it.
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Calculating money odds is easy. Count - or estimate - the size of the pot and
compare it to your investment. You also might want to estimate how much money
figures to be in the pot at the end of the hand, and compare that against your
expected investment. When money odds exceed drawing odds, you have the best of
it. Otherwise, you're drawing long.
(Exercise 10. Calculate the odds against making a flush when you hold two suited
cards, and two cards of your suit appear on the flop.)
Try working this out yourself. But if you are having difficulty, just read on.
Let's say you're holding A♥ J♥ and the flop is 8♥ 7♣ 3♥. Since there are thirteen cards of each suit in a deck, and you've accounted for four of them, only nine of the remaining 47 cards can be hearts. (Yes, there are three aces and three jacks that may also give you the best hand, but for this exercise we're only interested in learning how to calculate the chances of making a flush.)
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Like last issue's problem about the chances of flopping a set when you hold a
pair in your hand, it is also simpler to calculate the number of ways to miss
your flush, subtract the misses from the universe of possibilities, and the
result will be our answer.
On the flop there are 47 unknown cards. Since nine of them are hearts, the remaining 38 will not help you. If you miss your flush on the turn, there are now only 46 unknown cards. Since nine of them are hearts, 37 others won't help you. Let's multiply fractions. We've done this before, so it should be easy. Multiply the numerator of the first fraction by the numerator of the second, and perform the same calculation for the denominators. The result-to- 38/47 x 37/46 equals 1406/2162.
If you subtract the number of misses (1406) from the total number of possible events (2,162), you are left with 756 combinations that result in a flush. Now divide 756 by 2,162. The answer is 0.35 (or 35 percent). If you flop a four-flush, you'll make your flush 35 percent of the time.
Do you want to convert that percentage into odds? Here's how to go about it. Subtract 35 percent from 100 percent, and divide that by 35 percent (100 - 35 = 65; 65 / 35 = 1.86). The odds against completing your flush are 1.86-to-1. If the pot figures to pay 2-to-1 or more on your investment, it is a profitable draw in the long run - regardless of whether you make the flush now. Even if you were to miss the next twenty-seven times you're in this situation, drawing to your flush still has a positive expectation under these circumstances.
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Since you have no control over the cards that will be dealt, you can only focus
on making plays with positive expectations. That's what having the best of it is
all about. It's also why players can never win over any extended period at
craps, roulette, baccarat, and any of the other pit games, with the exception of
twenty-one.
Every professional poker player, and every skilled amateur who wins steadily at the game, takes the best of it most of the time. What separates winning players from the rest of the pack is this simple fact: Winning players take the gamble out of poker.
(Exercise 11. What are the odds against making a straight when you flop an open-ended straight draw?)
This is very similar to the flush problem. If you have difficulty with it, you ought to carefully reread each of the articles in this series. But if you can solve this problem effortlessly, you're well on your way to a passing grade in hold'em mathematics, and you are far ahead of any opponent who neither understands, nor can perform these calculations. But just in case you're having trouble, here's how to go about it.
If you hold 9♦ 8♦, and the flop is A♣ 7♠ 6♥, you've flopped an open-ended straight draw. Either a ten or a five completes your straight. Since there are four of each, any of these eight cards complete your hand.
Once again, we're going to multiply fractions. In this case you'll miss your straight 39 times out of 47 on the turn, and 38 times out of 46 on the river. When you multiply 39/47 x 38/46, you'll find that you'll miss your straight 1482 times out of 2162 attempts. By subtracting the number of misses from the universe of possibilities (2162 - 1482), you find that you will make a straight 680 times for every 2,162 times you flop an open-ended straight draw.
By dividing 2,162 into 680, you'll find that you'll complete a straight 31.5 percent of the time - which equates to odds of 2.17-to-1 against making your hand.
(Exercise 12. What are your chances of flopping at least one pair if you hold ace-king before the flop?)
Should you follow conventional wisdom and raise with A-K, or is it better to quietly call and plan to get aggressive only when the flop is favorable? This has been one of poker's ongoing strategy dilemmas ever since poker theorist Mike Caro suggested that calling might be a better play than power-raising whenever you've been dealt Big Slick. Before you can even think about taking a position on this issue, you need to know how often you figure to flop at least a pair with this holding. After all, while Big Slick might be a potent drawing hand, it must improve to offer anything more than bluffing value.
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If you've been able to work out all the problems in this series of articles, you
should probably be able to work this one out without having to read any further.
Once again, you'll follow these familiar procedures. Begin by computing the
number of ways you can miss with this hand - then subtract the number of misses
from the total number of possibilities to obtain your answer.
Since you already hold one ace and one king in your hand, there are only six of these cards among the remaining fifty, along with 44 cards that won't help you. Forty-four out of 50 times, you won't catch an ace or a king on the first card that flops. When the first card misses, there are still 6 good cards left in the deck, along with 43 of the remaining 49 that won't help you. If both the first and second card are neither ace nor king, six good cards and 42 bad ones comprise the 48 unknown cards left in the deck.
If you multiply 44/50 x 43/49 x 42/48, you'll come up with 79,464/117,600. When you subtract the 79,464 ways you can miss catching at least an ace or a king from the universe of 117,600 possible combinations, you'll find there are 38,136 ways to improve to at least a pair of aces or kings. Divide 117,600 into 38,136 shows that 32.4 percent of the time you'll catch at least an ace or king - which equates to odds of 2.1-to-1 against improving A-K on the flop. While that doesn't provide a definitive answer to the question of whether raising or calling is a better strategy, at least you have a few facts on hand when you ponder this problem.
Since you've diligently worked out all the problems in this series of lessons, here's some lagniappe for your efforts - a chart showing the chances of making your hand, along with the odds against making it, for situations ranging from 15 outs, where you are an odd-on favorite, to one out, where you're a decided long shot.
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Outs
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Chance of Success |
Odds Against Success. |
15 |
54.1% |
0.8-to-1 |
14 |
51.2% |
1.0-to-1 |
13 |
48.1% |
1.1-to-1 |
12 |
45.0% |
1.2-to-1 |
11 |
41.7% |
1.4-to-1 |
10 |
38.4% |
1.6-to-1 |
9 |
35.0% |
1.9-to-1 |
8 |
31.5% |
2.2-to-1 |
7 |
27.8% |
2.6-to-1 |
6 |
24.1% |
3.1-to-1 |
5 |
20.3% |
3.9-to-1 |
4 |
16.5% |
5.1-to-1 |
3 |
12.5% |
7.0-to-1 |
2 |
8.4% |
10.9-to-1 |
1 |
4.3% |
22.3-to-1 |
There's a lot of data in this chart, but I wouldn't bother memorizing most of
it. After all, how often are you going to stick around when you hold a hand with
just three outs? Nevertheless, you should memorize the percentages and odds for
9 outs (flush draw); 8 outs (open-ended straight draw); and four outs (two-pair
draw). Even if you can't compute the others in the heat of battle, you can
interpolate. Each additional out adds between 3 and 4 percentage points to your
chances. If you have 12 outs and know that a 9-out hand has a 35 percent chance
of improving, you won't be too far afield if you assume your chances of winning
are between 43 and 47 percent.
Some of this material may have been difficult to absorb, and if that's the case, keep studying until you understand how - and why - these calculations are performed. Even if you've been able to work through these problems, much of it will be forgotten unless you reread these articles periodically. Still, if you've come this far you certainly know more about this subject than most of your opponents. All you need to do now is put it into play. The results should soon be apparent in your win rate.
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