By poker_boy | 5 Comments
Last updated: Friday, July 22, 2011 | 92 Views Tags: aces club, diamond, four aces, mates, playing poker, probability, queen, royal flush, spade
We’ve had a reader question come in on the subject of poker. Let’s take a look:
In my hand I had two aces (Spade and Heart). My friend had a Jack (Clubs) and a 7 (irrelevant).
On the table there were another two aces (Club and Diamond). A King (Clubs). A Queen (Clubs) and a 10 (Clubs).
What are the chances?
The probability of getting a Royal Flush is ‘0.000154%’.
The probability of getting Four Aces is ‘0.0018%’.
Case 1 — you get AA and he gets 2 royal flush cards (other than an Ace).
————–
(4 C 2) * 2(4 C 2) * 43 = 6*12*43 = 3096 combos
Case 2 — you get AA and he gets 1 card to a Royal.
———
(4C2) * (8 * 46) = 2208 combos
Case 3 — he gets 2 cards to a Royal and you get 1 ace
———-
4(4C2) * 44 = 24 * 44 = 1056 combos
Total combos = 3096 + 2208 + 1056 = 6360
Out of a possible: (52 C 2) * (50 C 2) * (48 C 5)
Probability = 1 in 437,324,057 <——-
Now, that’s just the probability that YOU get quad aces and HE gets a royal. I suppose the chance of either one of you getting the quads and royal is twice that, 2 out of 437324057. As for any two players in the game (you said you and some of your friends), it would be a little higher.
I usually check my work using a 2nd method but I don’t feel like it now. I’m about 85-90% sure I didn’t mess up, for what that’s worth 
*Edit*
I’m pretty sure you can’t just multiply P(quad A) * P(royal), because they’re not independent probabilities.
Mike Smith
July 22, 2011
I’m not that good at math & statistics. Given that Royal Flush’s & 4 of a kind (outside of if you play with wilds or some weird game) are rare in and of themselves (even in 7 card games), seeing something like that has to be an extremely rare phenom.
July 22, 2011
The chances are soooo high that i don’t believe what you’re saying
July 22, 2011
0.000154 times 0.0018 = 2.77200 Ã 10-7
in simple english almost impossible
July 22, 2011
This actually happened in 2008 during the main event. It was one of the most talked about hands that year, so I’d be surprised if this wasn’t where you got the idea. The odds are 1 in 2.7 billion.
July 22, 2011
"in simple english almost impossible:
And yet it happened at the world series of poker