Probability hitting a Royal Straight Flush

[tolvan]

What is the probability of hitting a Royal Straight Flush?

How do you calculate a thing like that?

[elitegimp]

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What is the probability of hitting a Royal Straight Flush?

How do you calculate a thing like that?

[/ QUOTE ]

The probability depends on what game you are playing -- you are more likely to see a Royal in 7-stud than 5-stud (because each player gets more cards).

To calculate it, you count the number of hands that are royal flushes and divide by the the number of hands you can deal.

Example -- in 5-stud, there are 4 different Royal flushes (assuming order doesn't matter, so AKQJT is the same as TJQKA). There are 52 choose 5 = 2598960 possible ways to deal a 5-stud hand, so the probability of getting a royal is 4/2598960, or 1/649740 (0.00015%)

[tolvan]

Ok, what is the probability in Hold'em?

[elitegimp]

from here

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* If you have 2 Sooted Broadway cards, you can make a Royal if
(a) you hit the other 3 broadways of your suit, plus two random cards (there are 47*46/2 = 1081 different boards that look like that), or by having the board show a royal in one of the three other suits (3 different boards) => 1084 different boards will give you a royal. Further, there are C(50,5) = 2118760 unique boards given you hold two [specific] suited broadways, so you have a 1084/2118760 = 0.00051 chance (1/1954.58 or 1953.85 to 1) of hitting the royal.

* If you have two off-suit broadway cards, you can get a royal if you either hit a royal in either of your suits (need 4 other specific cards to show, plus one card can be anything) or if the board is a royal in one of the two suits you don't hold. So 2*1*46 =92 ways for you to play a card from your hand and have a royal, plus 2 ways for you to not play a card from your hand and have a royal => 94/2118760 = 0.0000437 chance (1/22540 or 22539 to 1).

* If you have only one broadway card in your hand, you can hit a royal in your suit 46 different ways (you need 4 specific cards, plus the 5th card can be anything left in the 46 card deck), or you need to hit one of the 3 royals that the board can show => 49/2118760 chance = 0.0000231 (1/43240 or 43239:1)

* If you have no broadway cards, then you're counting on the board to provide you a royal, and there are 4 ways for that to happen => 4/2118760 chance = 0.00000189 (1/529690 or 529689:1)

In general, if you have 2 random cards plus the board to make a royal with, you can treat it like a 7-card board. There are 4 sets of 5 cards to give you a royal, plus the remaining 2 cards can be anything => 4*47*46/2 = 4324 possible combinations and a total of C(52,7) = 133784560 ways to deal 7 cards => 4324/133784560 = 0.0000323 chance (1/30940 or 30939 to 1 against) that you hit a royal.

So to answer your question (I think) -- if you play 30940 hands, you expect to hit exactly one royal (but are guaranteed nothing, obviously).

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So one in 30940, not considering your hole cards (i.e. 2 random hole cards, 5 random cards on the board)