# Royal flush frequencies

#### loopmeister

##### Rock Star
Silver Level
I play on a very small site that has ~200 people seated at any given time on average.
They pay out a prize to the best hand of the day. I've noticed that almost every single day the best hand is a royal flush! Now 200 people seated means around 40 000 hands per day (all tables are 6 or 8 handed).

Is there something fishy going on here?
Is the site rigged?

Well, I did some math to confirm, and the result may surprise you:

- I assumed that every hand with RF potential goes to showdown with everyone calling with RF cards. This will make the results look better, but the math is much simpler. Maybe when I'm bored (er, I mean, more bored) I'll do a more realistic simulation.

- Empirically, my calculations suggest that your own chance of getting a Royal Flush are about 1 in 25 000. This includes seeing the RF on the board, or holding one of the cards in your hand. E.g. you have Qc 7s, and the board is Ac Jc Tc Kc 7d. The first case happens so rarely that it doesn't affect the results. The latter case will be influential if you insist that a RF includes both pocket cards.

- But with a table of 6, under the same assumptions, the odds improve dramatically to about 1 in 5600 hands!

- For 8 players, you should see a royal flush every 3900 hands, give or take!

So on a site that sees 40000 hands played per day, even a conservative approach of 6 seat tables sees the probability of seeing at least one RF on any given day is
p(RF) = 1 - p(no RF)
= 1 - (5599/5600)^40000
= 99.9%
Even a slow day, with only 20 000 hands played, the chance of having a RF appear somewhere is 97.2%

Since I've gone this far, I've attached a plot of the probability of seeing a Royal flush on a 8-handed tables after watching n games.

This surprised me, but combinatorial problems often do.

#### Attachments

• RoyalFlush.jpg
19.6 KB · Views: 70

#### loopmeister

##### Rock Star
Silver Level
The results above were based on simulations of 1 million poker hands. The odds of getting a royal flush are small enough that there is significant variance in the results between runs, even with such a large sample size.
I've subsequently worked out the exact odds of getting a royal flush in texas holdem: 0.0032% or 1:30940

I'm sure many of you have played well over 31 thousand hands of online poker - if so you would most likely have already received this hand. By the time you've played 93 000 hands you almost certainly (95% probability) would have received a royal flush (assuming you didn't fold that AT suited )

#### Jack Daniels

##### Charcoal Mellowed
Silver Level
Wow, and I thought I was a math geek. Nice work Loop. I enjoyed reading all of it. Now I have to go back and see how many hands I have on record as I've had two Royal Flushes online (and one live, but that's out of scope). Interesting stuff.

Silver Level

#### F Paulsson

##### euro love
Silver Level
For the sake of being a nit-picking bastard, I just want to point one thing out:

For 8 players, you should see a royal flush every 3900 hands, give or take!
This presumes that every hand is dealt to the river and that every person around the table goes all the way to showdown. I know that you realize this, loopmeister, but I wanted to add it for stringency. The probability of seeing a royal flush at your table goes down drastically because of this, but not so far down that there couldn't be one every day on a small site.

/FP

#### loopmeister

##### Rock Star
Silver Level
For the sake of being a nit-picking bastard, I just want to point one thing out:

This presumes that every hand is dealt to the river and that every person around the table goes all the way to showdown. I know that you realize this, loopmeister, but I wanted to add it for stringency. The probability of seeing a royal flush at your table goes down drastically because of this, but not so far down that there couldn't be one every day on a small site.

/FP

Precisely. I think I did mention it explicitly.

I imagine on a site like partypoker there must be a royal flush every hour or so.

Silver Level