royal flush probability
July 22nd, 2015 at 5:44:44 PM
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What are the odds of a royal on a 100 multi-hand jacks or better game? You only get one flop initially then draw a 100 hands on this version, not sure if they are all like that.
July 22nd, 2015 at 6:53:59 PM
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I have a feeling I'm making a mistake posting this without checking it through simulation first, but I get a probability of 1 in 274.564, assuming you always play a "royal or nothing" strategy (for example, if you are dealt A-A-A-A-2, keep one of the aces and discard the other four cards; if you are dealt a suited J-10-9-8-7, keep the jack and 10; if you are dealt 9-9-9-9-6 or suited 9-8-7-6-5, discard all five cards).
You keep the highest number (so, for example, in 3-1, you keep the 3 cards of the same suit; in 2-2-1, you keep either of the sets of 2 cards of the same suit).
If you have five suited high cards, obviously that's a Royal Flush - in fact, 100 of them - and the probability is 1.
If you have 4, you have a 1/47 chance of getting the fifth card, which means you have a 46/47 chance of not getting it, so you have a (46/47)100 chance of not getting any royals in 100 of these hands, and a 1 - (46/47)100 chance of getting at least one.
Similarly, with 3 suited high cards, you have a 1 - (1 - 2 / (46 x 47) )100 chance of at least one royal;
with 2, a 1 - (1 - 6 / (47 x 46 x 45) )100 chance;
with 1, a 1 - (1 - 24 / (47 x 46 x 45 x 44) )100 chance;
with none, a 1 - (1 - 120 / (47 x 46 x 45 x 44 x 43) )100 chance.
Multiply the numbers of each hand by the appropriate probability, add them together, and divide by 2598960.
| High cards by suit | Number of hands |
|---|---|
| 5 | 4 |
| 4 | 4 x 5 x 32 |
| 4-1 | 4 x 5 x 15 |
| 3 | 4 x 10 x (32 x 31 / 2) |
| 3-1 | 4 x 10 x 3 x 5 x 32 |
| 3-2 | 4 x 10 x 3 x 10 |
| 3-1-1 | 4 x 10 x 3 x 25 |
| 2 | 4 x 10 x (32 x 31 x 30) / 6 |
| 2-1 | 4 x 10 x 3 x 5 x (32 x 31) / 2 |
| 2-1-1 | 4 x 10 x 3 x 25 x 32 |
| 2-1-1-1 | 4 x 10 x 125 |
| 2-2 | 6 x 100 x 32 |
| 2-2-1 | 6 x 100 x 2 x 5 |
| 1 | 4 x 5 x (32 x 31 x 30 x 29) / 24 |
| 1-1 | 6 x 25 x (32 x 31 x 30) / 6 |
| 1-1-1 | 4 x 125 x (32 x 31) / 2 |
| 1-1-1-1 | 625 x 32 |
You keep the highest number (so, for example, in 3-1, you keep the 3 cards of the same suit; in 2-2-1, you keep either of the sets of 2 cards of the same suit).
If you have five suited high cards, obviously that's a Royal Flush - in fact, 100 of them - and the probability is 1.
If you have 4, you have a 1/47 chance of getting the fifth card, which means you have a 46/47 chance of not getting it, so you have a (46/47)100 chance of not getting any royals in 100 of these hands, and a 1 - (46/47)100 chance of getting at least one.
Similarly, with 3 suited high cards, you have a 1 - (1 - 2 / (46 x 47) )100 chance of at least one royal;
with 2, a 1 - (1 - 6 / (47 x 46 x 45) )100 chance;
with 1, a 1 - (1 - 24 / (47 x 46 x 45 x 44) )100 chance;
with none, a 1 - (1 - 120 / (47 x 46 x 45 x 44 x 43) )100 chance.
Multiply the numbers of each hand by the appropriate probability, add them together, and divide by 2598960.
