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 |
Pair of Jacks or better |
Pair of aces or better |
Any two pairs or better |
Three of a kind or better |
Straight or better |
Flush or better |
Full house or better |
Four of a kind or better |
Straight flush |
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.