Royal flush frequencies
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
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.