Royal on the Flop

[Sparks]

In a 15-30 hold 'em game at Hawaiian Gardens last night a guy at my table flopped a royal flush. What are the odds of a player doing that?

If I play an average of 4 hours per day, with say 7 other players taking hands at my table, how long would it be before I should expect to see it again?

Thanks.

Sparks

[Marm]

Since its 5 specific cards, then the same for a royal in Draw. approx 2.6 million - 1. (Off the top of my head)

The actual number should be 4(52!)/(5!(47!)) correct?

[Piz0wn0reD!!!!!!]

[ QUOTE ]
Since its 5 specific cards, then the same for a royal in Draw. approx 2.6 million - 1. (Off the top of my head)

The actual number should be 4(52!)/(5!(47!)) correct?

[/ QUOTE ]

the first card is not specific in suit.

[elitegimp]

[ QUOTE ]
Since its 5 specific cards, then the same for a royal in Draw. approx 2.6 million - 1. (Off the top of my head)

The actual number should be 4(52!)/(5!(47!)) correct?

[/ QUOTE ]

Close... (52!)/(5!47!) = number of possible 5 card hands... only 4 of them are royal flushes, so it's 4/((52!)/(5!47!)) = 4*5!*47!/52!, or 1/649740.

I'm still not totally clear on the math for including multiple hands pre-flop, so I'm not gonna try to tackle the odds of someone flopping it at a 7-person table or whatever (hopefully someone else will /images/graemlins/smile.gif)

to the original poster:
[ QUOTE ]

If I play an average of 4 hours per day, with say 7 other players taking hands at my table, how long would it be before I should expect to see it again?

[/ QUOTE ]
That will depend on how many hands you play in an hour /images/graemlins/smile.gif I believe the typical number is 60 hands / hr / table online, and 40 hands / hr live. Do those seem reasonable?

[gaming_mouse]

[ QUOTE ]
I'm still not totally clear on the math for including multiple hands pre-flop, so I'm not gonna try to tackle the odds of someone flopping

[/ QUOTE ]

Since it is impossible for two players to both flop a royal flush, we need only calculate the chance of a specific player flopping one and then multiply that by 7. You have already correctly calcualted the chance that a specific player flops a royal as 1 in 649740, so the chance that 1 of 7 players flops one is 7 in 649740, or:

1 in 92820

Now, to answer the OP's other question, in a live game you will play about 35 hands/hour, or about 140 hands/4 hour session. After 663 such sessions, the chance that you will have seen another flopped royal is about 63%:

1 - ((92 819 / 92 820)^92 820) = 0.632122541

Note, also, that this is almost exactly 1-1/e, where e is the base of the natural logarithm.

Anyway, if you are playing 4 hours a day every day, expect to see this every couple years or so.

gm

[Thythe]

If we wanted to get extremely picky wouldn't we have to calculate some probabilities of actually seeing the flop when you hold 2 to the royal. For example, I raise AKs preflop and get 5 callers. Most of the callers aren't going to have 72o or hands like that. It is more likely they have QQ, JT, JJ, etc. It is also very probable that they have cards to your royal if they called in the first place. Clearly this would be almost impossible to do any type of math on, though...

[gaming_mouse]

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If we wanted to get extremely picky wouldn't we have to calculate some probabilities of actually seeing the flop when you hold 2 to the royal.

[/ QUOTE ]

Yes, good point, but it is not unreasonable to assume that suited broadways are always played. Against a raise or 3-cold people will sometimes fold them, but the assumption is still reasonable.

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For example, I raise AKs preflop and get 5 callers. Most of the callers aren't going to have 72o or hands like that. It is more likely they have QQ, JT, JJ, etc. It is also very probable that they have cards to your royal if they called in the first place.

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None of this matters. The only thing we have to worry about is when people fold their suited broadways. The chance that other people may hold the cards to complete your royal is implicitly factored into my calculation. So this type of thing is not impossible to do the math on. The only thing that is impossible to estimate exactly is the chance of people folding suited broadways.

gm

[pzhon]

You also have to worry about not seeing a flop with suited broadway cards because you raised and everyone folded.

[Thythe]

Yeah I just realized my mistake. It certainly is used in the calculation.

[gaming_mouse]

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You also have to worry about not seeing a flop with suited broadway cards because you raised and everyone folded.

[/ QUOTE ]

Yes. Nice point, pzhon. Anyway, the calculation I gave should give a good approximation nonetheless -- even ignoring times you fold and times you fold out everyone else. And we can use it as a lower bound on the odds.

gm