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ABOUT ODDS, PERMUTATIONS AND RETURN ON INVESTMENT


WHAT DOES THE TERM CARD ODDS EXACTLY MEAN?

If you have not already noticed, probability is an immense factor in Texas Hold 'em. E.g., there are 2,598,960 possible hands in a 52­card deck but only 4 Royal Flushes. If the ordinary serious poker player is dealt out 100,000 hands in their lifespan, they will never hold (on the first five cards) more than 4 percent of all the possible hands. And in all likelihood a lot less.

Working out straight card combinings for the purpose of this text are called Card Odds (you will be introduced to additional kinds of odds later). Card Odds can expose a lot of quite interesting data.

For instance, how many pat straight flushes will you see in your lifetime? To find out that number, the likely amount of hands that could be dealt during your lifetime is counted on by the following computation: 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 approximation, as most people will never come close to this number of complete hands in Texas Holdem. Based on this level of play, the number of pat (on the first five cards) poker hands that you should get during your lifespan is calculated from the card odds and tabularized 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

So statistically, you should see a pat straight flush on your first five cards once or twice during your lifetime. Most intermediate poker players will never see even one. Card players often talk of having a ‘hot streak’ or a ‘run’. Mathematically, ‘stripes’ do not exist. But imagine you did have an astonishing run of cards one evening. What would the odds be of having five back-to-back 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 would have to study those cards in the dark though, because our sun will be long gone by that time.

Up to now we know that the real Rounders use card odds to make playing decisions. A determination made without taking into account card odds makes poker a hazarding game. The probabilities of finishing a flush or a straight, the probability of getting an over card (face card), the percentage of times you are going to flop a card to match your pocket pair - are all highly significant factors in Texas Hold’em. Knowledge of these statistics is key to winning.

Here are some other more common 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%

Chance 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%

Chance you will hit a pair at the flop, holding two unpaired cards in the pocket: 32.4%

Probability of being dealt AA: .45%

Probability of no one holding a particular card, by amount of players, presuming you do not have that card, by number of overall 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%

Chance someone else does not have an ace, assuming you do hold an ace, by total number of players:

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?


Let’s consider the good example of having 4 outs (four cards you need to

make your hand). Suppose you are holding 6c 7d and the flop comes 9s 10h Kc.

In this example you need an 8 to make the straight. Since there are four 8’s in the deck of cards, you have 4 outs.

ODDS WITH ONE CARD TO COME


Calculating the odds with one card to come is comparatively straightforward.

When you are anticipating drawing to 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 do not build your hand and four do. 42:4 or 10.5:1 = about 9%. We opt to use the percentage as it assists when calculating Pot Odds (to come later).

ODDS WITH TWO TO COME


To estimate the appropriate odds with two cards to come, you must first find out the total number of two-card combinations getable after the flop. The easiest formula to calculate this is by multiplying the amount 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 estimate 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, on that point 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.

EIGHTS ON THE TURN OR RIVER


If an eight comes on the turn, there are 46 unseen cards remaining. But you're no more interested in the three remaining eights, so you can deduct 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 MATH


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%.


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