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The mathematics of Poker
Poker is a game of probability, not certainty (otherwise, there would be no betting). There are many books out there that dissect the mathematics of poker. While you can probably leave your slid rule and calculator at home, to be a successful poker player you should have an appreciation of the numbers.
Every deck of fifty-two cards has 2,598,960 possible five-card poker hands. Below are the odds of being dealt the following hands with the first five cards.
| TYPE OF HAND | QTY IN DECK | ODDS AGANIST |
| Royal Flush | 4 | 649,739:1 |
| Straight Flush | 40 | 64,973:1 |
| 4 of a Kind | 624 | 4,164:1 |
| Full House | 3,744 | 693:1 |
| Flush | 5,108 | 508:1 |
| Straight | 10,200 | 254:1 |
| 3 of a Kind | 54,912 | 46:1 |
| 2 pairs | 123,552 | 20:1 |
| 1 Pair | 1,098,240 | 1.25:1 |
| No Pair | 1,302,540 | Evens |
Before trying to fill that inside straight (it will happen only once in eleven tries), you should have a good idea of what your chances are. Below are the odds against improving your hand in draw they can also help you think about your poker odds in stud games as well:
| CARDS | CARDS | IMPROVED | ODDS |
| KEPT: | DRAWN: | HANDS: | AGAINST: |
| 1 Pair | 3 | 2 Pairs | 5.25:1 |
| 1 Pair | 3 | 3 of a Kind | 8:1 |
| 1 Pair | 3 | Full House | 97:1 |
| 1 Pair | 3 | 4 of a Kind | 359:1 |
| 1 Pair | 3 | Any Improvement | 2.5:1 |
| 2 Pair | 1 | Full House | 11:1 |
| 3 of a Kind | 2 | Full House | 15.5:1 |
| 3 of a Kind | 2 | 4 of a Kind | 22.5:1 |
| 4-Straight | 1 | Straight | 5:1 |
| (open-ended) | |||
| 4-Straight | 1 | Straight | 11:1 |
| (inside) | |||
| Four-flush | 1 | Flush | 4.5:1 |
| Three-flush | 2 | Flush | 23:1 |
| Two-flush | 3 | Flush | 96:1 |