Introduction
Development
Introducing Poker
Ranking Hands
Chips
The sequence of play
Shuffling
Dealing
Betting Interval
Betting Small and Big Blinds
Table stakes
Using wild cards
Probability of holding
First betting interval
Strategy-1
Strategy-2
Seven-Card Stud
Other forms of poker
Texas Hold'em Basic Hand
Five - Six card Omaha
Poker Sense
Slow Playing
Other Gambling Card Game
Blackjack
Brag
Seven-card Brag
Gin Rummy
Glossary

The arithmetic of draw poker

In table 1, the number of ranking poker hands, and how often they are likely to be dealt, were shown. From these you can estimate the value of any hand as dealt. For example, almost exactly half of all hands will include at least one pair. However, a player will not receive a hand better than this more than roughly 7.5 per cent of the time, or about once in 13 hands. Table 3 gives the chances of being dealt each hand or better.

TABLE 3: Probability of holding any particular hand or better in five cards dealt
Any pair of better 1 in 2 deals	Pair of Jacks or better 1 in 5 deals
Pair of Aces or better 1 in 9 deals	Any two pairs or better 1 in 13 deals
Three of a kind or better 1 in 35 deals	Straight or better 1 in 132 deals
Flush or better 1 in 273 deals	Full house or better 1 in 590 deals
Four of a kind or better 1 in 3,914 deals	Straight flush 1 in 64,974 deals

TABLE 4: Probability in percentages of holding the best hand at the deal
	Number of players
	2	3	4	5	6	7	8
Pair of Jacks	79%	63%	50%	40%	32%	25%	20%
Pair of Aces	89%	79%	70%	62%	55%	49%	43%
Two pairs	93%	86%	80%	74%	68%	63%	59%
Three of a kind	98%	94%	92%	89%	87%	84%	82%

Table 5: Probability of improving one pair by drawing three cards
Odds against any improvement	2.4 to 1
Odds against two pairs	4.8 to 1
Odds against three of a kind	7.7 to 1
Odds against full house	97 to 1
Odds against four of a kind	359 to 1

Table 6: Probability of improving one pair by drawing two cards
Odds against any improvement	2.8 to 1
Odds against two pairs	4.8 to 1
Odds against three of a kind	11.9 to 1
Odds against full house	119 to 1
Odds against four of a kind	1080 to 1

If being dealt two pairs is to receive a good hand, in that you will equal or beat it only once in 13 times, how likely is it that you will be holding the best hand if you are one of six players? The answer is approximately 68 per cent of the time, or odds of slightly better than 2 to 1 on that you hold the best hand.
Table 4 gives an indication of how good the hand you are dealt with is; up of three of a kind, in terms of how likely it is to be the best hand dealt.
From this table it can be seen that if you are dealt three of a kind, the chance of you holding the best hand at that stage is 82 per cent (better than 4 to 1 on) even if there are eight players in the game.

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