37 EVERYTHING YOU ALWAYS WANTED TO KNOW (AND MAYBE DIDN�T) ABOUT ODDS, PERMUTATIONS AND RETURN ON INVESTMENT WHAT ARE CARD ODDS? If you haven�t already noticed, probability is a huge factor in Texas Hold 'em. For example, there are 2,598,960 possible hands in a 52- card deck but only 4 Royal Flushes. If the average serious poker player is dealt 100,000 hands in their lifetime, they will never hold 38 (on the first five cards) more than 4 percent of all the possible hands. And likely a lot less.
Figuring out straight card combinations for the purpose of this text are called Card Odds (you will be introduced to other kinds of odds later). Card Odds can reveal some quite interesting information.
For example, how many pat Straight Flushes will you see in your lifetime? To determine that number, the expected number of hands that could be dealt during your lifetime is estimated by the following calculation: 10 complete poker hands / hr. x 5 hrs. / game x 50 games / yr.
x 40 yrs. / poker life = l00,000 hands of poker per lifetime.
This is a pretty aggressive estimate, as most people will never come close to this number of Complete Hands in Texas Hold�em.
Based on this level of play, the number of pat (on the first five cards) poker hands that you should get during your lifetime is calculated from the card odds and tabulated as follows: Cards Dealt Number of Pat Hands No pair 50,000 One pair 40,00 Two Pair 5,000 Three Of A Kind 2,000 Straight 400 Flush 200 Full House 170 Four Of A Kind 25 Straight Flush 1.4 Royal Straight Flush 0.15 39 So statistically, you should see a pat Straight Flush on your first five cards once or twice during your lifetime. Most average poker players will never see even one.
Card players often talk about having a �lucky streak� or a �run�.
Mathematically, �streaks� don�t exist. But suppose you did have an amazing run of cards one evening. What would the odds be of having five consecutive Straight Flushes in a row? Royal Flush 4 .0000015391 Other Straight Flush 36 .0000138517 Four Of A Kind 624 .0002400960 Full House 3,744 .0014405762 Flush 5,108 .0019654015 Straight 10,200 .0039246468 Three of a kind 54,912 .0211284514 Two Pairs 123,552 .0475390156 One Pair 1,098,240 .4225690276 Nothing 1,302,540 .5011773940 Total 2,598,960 1.0000000000 In every 1.7x1024 deals . . . or once in every 700,000,000,000,000,000,000 years. You�d have to read those cards in the dark though, because our sun will be long gone by that time.
Players use card odds to make playing decisions. A decision made without taking into account card odds makes poker a guessing game. The chances of finishing a flush or a straight, the probability of getting an over card (face card), the percentage of times you're going to flop a card to match your pocket pair - are all extremely important factors in Texas Hold�em.
40 Knowledge of these statistics is key to winning.
Here are some other basic probabilities that you should know about: � You need one more heart to make your flush on the turn or river - 35% � Probability of hitting an open-ended straight draw (i.e. 4 straight cards, need one on either end to hit on turn or river)- 31.5% � Probability of being dealt suited cards: 23.5% � Probability of hitting a three or Four Of A Kind at the flop when you hold a pocket pair: 11.8% � Probability you will make a pair at the flop, holding two unpaired cards in the hole: 32.4% � Probability of being dealt AA: .45% � Probability of no one holding a specific card, by number of players, assuming you do not have that card, by number of total players.
2 - 84.5% 3 - 70.9% 4 - 59% 5 - 48.6% 6 - 39.7% 7 - 32.1% 8 - 25.6% 9 - 20.1% 10 -15.6% � Probability someone else does not have an ace, assuming you do have an ace, by total number of players: 41 2 - 88.2% 3 - 77.5% 4 - 67.6% 5 - 58.6% 6 - 50.4% 7 - 43% 8 - 36.4% 9 - 30.5% 10 - 25.3% HOW ARE THE ODDS CALCULATED? Lets look at the example of having 4 outs (four cards you need to make your hand). Say you're holding 6c 7d and the flop comes 9s 10h Kc. In this case you need an 8 to make the straight. Since there are four 8�s in the deck, you have 4 outs.
YOUR POCKET THE FLOP 42 ODDS WITH ONE CARD TO COME Calculating the odds with one card to come is relatively straightforward.
When you're looking to make the Inside Straight, you have four outs. There are a total of 46 unknown cards (52 minus the 2 cards in your hand minus the 3 cards for the flop and the 1 turn card). 42 of the cards don't make your hand and four do. 42:4 or 10.5:1 = about 9%. I prefer to use the percentage as it helps when calculating Pot Odds (to come later).
ODDS WITH TWO TO COME To calculate the appropriate odds with two cards to come, you must first determine the total number of two-card combinations possible after the flop.
The easiest way to calculate this is by multiplying the number of cards available for the turn (47) by the number of cards available for the river (46) and dividing that number by 2 (because a card can't match itself). 47*46/2 = 1081.
A certain number of these 1081 two-card combinations will have eights in them. To determine odds properly, you need to calculate two more figures.
EIGHTS ON BOTH THE TURN AND THE RIVER One of the four eights can appear on the turn. And if one does, there will be three left for the river. If you multiply 4 by 3 and divide by 2 (because a card can't match itself) you see that there are six unique pairing of 8s.
43 EIGHTS ON THE TURN OR RIVER If an eight comes on the turn, there are 46 unseen cards remaining.
But you're no longer interested in the three remaining eights, so you can subtract those. This leaves 43 unseen cards that will make a unique pair with one of the eights. Multiply 4 (the number of 8s in the deck) by 43 (the number of unseen cards) to arrive at 172.
FINISH THE CALCULATION 172 plus 6 comes to 178 -- the total number of two-card combination that have at least one eight in them and as many as two eights.
Out of 1081 possible two-card combinations on the turn and river, 178 of those combinations help us make our hand. Subtract 178 from 1081 to find the number of combinations that don't make the straight (1081-178=903).
The odds against making a straight by the river are: 903:178, or 20%.
What About The Cards The Other Players Are Holding? Ever wonder why we never factor the opponents' cards or the Burn Cards when figuring out how many cards are left? The reason is that we only consider ?unseen cards?. If you saw what the Burn Cards were, or an opponent showed you his hand, you would know that those cards are not going to be drawn and could use that. We typically do not know what they have, so we don't even think about it when talking about odds.
44 For instance, take a standard deck of 52 cards, remove 2 Aces and burn 25 of them. If you drew the next card, what are the chances of it being an Ace? It would be 2/50 (2 Aces left out of 50 unseen cards). It would NOT be 2/25 just because you burned half the deck. Okay, do the same thing again, but this time you get to look at the Burn Cards. Let's say that of all the cards you burned, none were an ace. Now your odds are 2/25 because there are still 2 Aces and now only 25 ?unseen cards?.
You will find that you can easily remember a few of the most common situations for outs such as the four flush or straight draw but there has to be an easier way than memorizing the figures for every number of outs. The good news is that there is a way to get a good estimation of the odds without the heavy math and you can also use handy odds charts.
WHAT HANDS WILL WIN THE POT? The following are the most valuable starting hands in Texas Hold'em. This chart assumes a medium to loose $5-10 Texas Hold'em game. The results are based on a computer simulation of 5,000,000 played hands. The percentage shown indicates how many times in typical game that these hands win the pot.
2 Pair 31% Pair 27% Three Of A Kind 12% Straight 9% Flush 9% Full House 9% Bust (nothing) 2% Four Of A Kind 1% Straight Flush <1% Royal Flush <1%