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Poker Probabilities

I said before that the higher mathematics of poker are not very important. It doesn't help a player much to know the chances of being dealt a straight flush or a full house or even a pair. Yet most of the published tables of poker probabilities are confined to that kind of information.

It does help to know the odds against improving any particular hand. A summary of these odds is given on below. Memorize those odds. At the very least they will tell you when the pot is offering you good odds on a speculative play.

In the following discussion I give some special phases of poker mathematics.

Possible Poker Hands in a 52-Card Deck:

Straight Flush 40
Four of a Kind 624
Full House 3,744
Flush 5,108
Straight 10,200
Three of a Kind 54,912
Two Pairs 123,552
One Pair 1,098,240
No Pair, less than above 1,302,540
Total 2,598,960

Possible Hands of Less Value Than One Pair:

If Ace Counts High:
If Ace Counts Low:

Ace High
King High 502,860
King High
Queen High 335,580
Queen High 
Jack High 213,180
Jack High Ten High 127,500
Ten High Nine High 70,380
Nine High Eight High 34,680
Eight High Seven High 14,280
Seven High Six High 4,080



Total

1,302,540

Chances of Being Dealt Poker Combinations in the Original
Five Cards

Royal Flush 1 in 649,740
Straight Flush 1 in 64,974
Four of a Kind 1 in 4,165
Full House 1 in 694
Flush 1 in 509
Straight 1 in 255
Three of a Kind 1 in 47
Two Pairs 1 in 21
One Pair 1 in 2 1/2
No Pair 1 in 2

Blind-Opening Draw Poker

Chance of beating one opponent who has not bet voluntarily (that is, chance of beating the blind raiser when you are the blind opener:)

If you draw 5 cards 9 to 1 against you
If you draw 4 cards to an ace 1 1/2  to 1 against you
If you draw 3 cards to an A-K 2 to 1 against you
If you draw 2 cards to a flush 8 to 1 against you
If you draw 2 cards to a straight 8 to 1 against you
If you draw 2 cards to a straight flush 6 to 1 against you
If you draw 3 cards to a low pair (4, 3, 2) Even
If you draw 3 cards to a medium pair (5 to 9) 11/2 to 1 for you

On any better hand than a medium pair you should raise.

Comment: The pot usually offers 4 to 1. It will be seen that the usual plays stand to lose. However, the figures are complicated by the high cards in the hand that may pair.

The same principles apply to straight draw poker when you are next-to-last man and everyone but the dealer is out.

Every additional opponent reduces the odds in your favor on a particular hand. But every additional opponent has to put his money into the pot, thus increasing the odds you are getting for your own money. All poker calculations come to the same thing, though in varying degree: If you had the high hand to begin with, you figure to win. The more players there are against you, the fewer pots you will win but the more you will win each time you win the pot, and in the long run you still show a profit.

Chances of Improving in the Draw (Draw Poker)

Draws to One Pair

Drawing three cards to one pair—odds against making:

Two pair or better 2 1/2 to 1 against
Aces up or better
6 to 1 against
Three of a kind 8 to 1 against
Full house 97 to 1 against
Four of a kind 359 to 1 against

Drawing two cards to one pair and an ace kicker-odds against making:

Two pair or better 3 to 1 against
Aces up or better 4 to 1 against
Three of a kind
12 to 1 against
Full house 119 to 1 against
Four of a kind 1,080 to 1 against

Comment: It will be seen that if the player is sure he must have aces up or better to win, his chance is slightly better by holding the Ace kicker.

Draws to Three of a Kind

Drawing two cards to three of a kind—odds against making:

Full house or better 82/3 to 1 against
Full house 151/2 to 1 against
Four of a kind 22 1/2 to 1 against

Drawing one card to three of a kind and any kicker—odds against making:

Full house or better 103/4 to 1 against
Full   house 
14 2/3  to 1 against
Four of a kind
46 to 1 against

Comment: Holding a kicker to three of a kind decreases the chance of improvement and obviously can be advisable only for deceptive or tactical purposes.

Other Odds

The odds are 38 to 9 or to 1 against filling a fourflush; 39 to 8 or almost 5 to 1 against filling a double-ended straight; 43  to 4 or  to  1  against filling an inside or one-ended straight; 43 to 4 or to 1 against filling a full house when drawing to two pair.

With a double-ended straight-flush draw (such as the odds are 2 to  1  against making a straight or better,
to 1 against making a straight flush, 5 to 1 against making a flush, to 1 against making a straight.

With a one-ended straight-flush draw (such as or 8-7-5-4), the odds are 3 to 1 against making a straight or better, 46 to 1 against making a straight flush, 5 to 1 against making a flush, 14 to 1 against making a straight.
Drawing four cards to an ace, the odds are 21/2 to 1 against making a pair of aces or better. (In stud poker the odds are 31/2 to 1 against pairing an ace if no other ace shows.) Drawing two cards to A-K, the odds are 21/2 to 1 against making aces or kings, or better.

It is 23 to 1 against making a flush on a two-card draw and 22 to 1 against making a straight on a two-card draw (both ends open twice, as 10-9-8). It is 11 to 1 against making a straight or better on a two-card straight-flush draw such as 10-9-8.

Lowball

The following odds apply to lowball played with the bug. In each case it is assumed that the bug is not in the hand.

One card to 7-high 31/2 to 1 against
One card to 6-high
51/2 to 1 against
One card to 5-high 9 to 1 against
Two cards to any three cards under 7, to make 7-high or better 271/2 to 1 against
One card to 8-high 2 to 1 against
One card to 9-high Even

Comment: The more players there are in the pot, the more low cards can be assumed to have been dealt originally and the worse the chances are for improvement. The same fact decreases the danger of pairing, but this does not alter the fact that the chances of improvement are lessened.
 
Copyright 2006 - 2013 Pokeroligist.com. Content by Albert H. Morehead