That was the odds per ESPN commentators on WSOP main event hand (Day 1). Quad Aces (two aces in hole) lost to a Royal Flush. I'm sure the odds are exceptionally high but 2.7 billion to 1 sounds pretty crazy. Anyone good at figuring the precise odds here.

nevermind

Well this is my first attempt at this, but here's what I got.

I believe the overall odds are 1 in 978.8 Million. That is assuming it happens the way it did in the WSOP (pocket aces and pocket royal flush cards), the quads having only 1 ace, OR the RF having only one hole card to it.

To answer your post more specifically...

Assuming quad aces with two in the hole, BUT the RF had either 1 OR 2 in the hole, the odds would be 1 in 1.646 Billion.

And finally, the odds of both players having both cards to their hand in the pocket, 1 in 4.39 Billion.

OK - someone please tell me why I'm wrong . I'm not sure if the number of opponents matters, it would definately have to icrease the odds of 2 players starting with the 2 hole cards to make this happen though. My numbers assume heads up (if it makes a difference).

I also am not sure if it matters if player 1 has quads and player 2 has a RF OR if player 1 has a RF and player 2 has quads...but I don't think it does.

Here is my take on it

All possible starting hands -> C(52,2) = 1326
Player A holding AA -> P(A) = 6/1326 = 1/221
Player B holding KQ or KJ or KT or QJ or QT or JT (different suit than AA) -> P(B) = 12/1326 = 2/221

All possible remaining card combinations on board -> C(48,5) = 1,712,304
Boards with AAAA vs royal flush have four of the cards completely determined. The fifth card can be any of 44 cards.
So we have 44 good boards for our AAAA vs royal flush -> P(C) = 44/1712304 = 1/38916

For heads up game, the probability of AAAA vs. royal flush is P(A)*P(B)*P(C) = 1.05/(10^9)

For 10 players
Any of them holding AA -> P(D) = 1 - (1-1/221)^10 = 0.0443
Any of them holding KQ or KJ or KT or QJ or QT or JT -> P(E) = 1 - (1-2/221)^10 = 0.0869
P(D)*P(E)*P(C) = 9,9/(10^8)

I'm not sure though

I think for the KJ, etc. you need to use C(50,2) since player A has 2 aces.

One thing I did wrong and realized later last night was I didn't take into account that when player 1 has only 1 Ace in the pocket, that means player 2 can now have KQ, KJ, etc. of any of the other 3 suits, I was still using 2. Changing it to 3 gives me overall odds of 1 in 813.8M. Still probably wrong though

For P(B) you need to divide by C(50,2) = 1225 instead of 1326 to take player A's cards into account. Then you should multiply all of this by 2 so that player A can have the royal and player B the AAAA. This gives the correct answer 2.27 * 10-9 or about 439 million to 1. There is no way that 2.7 billion to 1 is correct for this problem. Perhaps they said 2.27 in 1 billion?

The most important thing to know about this type of problem is that you can get the EXACT answer for 10 players by simply multiplying your answer for 2 players by the number of ways to choose 2 players C(10,2) = 45. This is because only 2 players can have these 2 hands, so the pairs of players are mutually exclusive.

Great explanation Bruce. Thanks.

Here is the video of the hand
http://www.youtube.com/watch?v=XunAlp2azhA

You get 2.7 billion to 1 if you insist the royal flush have two specific ranks, say KQ.

So is it correct that the odds of being the Royal Flush hand, with both cards in the pocket, is about (439 million x 10) to 1? I had one of these a few weeks ago where I was the RF over AAAA - both cards in the pocket.



I've played about 30k SNGs - averaging something like 25-30 hands / SNG (I should look that up). So lets say 900k chances, but I probably would have folded at least half of the time I had suited broadway. (SNG = nitfest early or facing a raise.) So effectively I got a royal over AA in about 450k chances.

In close to 4.5 billion to 1 odds. So about a 1 in 10k chance I would have had this happen in by now, in 30k sngs. Or what the same odds as me winning the Sunday mil and making it deep another time (assuming some edge here)? Sweet. At least I took 2nd in a $78 SNG.

On any given hand, be it the first you ever played or the one last Thursday, the odds of quads aces vs. RF are 2.7 billion:1. You can't equate the expectation of it happening at any given to the odds that it's happened to you in x number of hands.

That's the number the show gave, and as Aaron pointed out, it is only the odds when the royal has particular hole cards like KJ, and actually that would be closer to 2.6 billion to 1, so I'm not so sure even that is what the show was calculating. For any hole cards, it is 439 million-to-1 as I computed in this thread. Now both of these are the odds for you holding either the royal or the quads. The probability is half (odds twice as long) for you holding the royal only.

Ok, so the odds of me holding the Royal (and both cards playing in each hand) is about 878 million to one.

Now of course I can calculate the probability of something happening with that odds in X number of chances.

The probability is the odds that it will happen on any of those occasions minus the odds it will happen multiple times. Px + Py - (Px * Py). But extrapolated out 450k times.

Although we can easily decide that the odds of it happening more than once are so negligible, that we just add up the probabilities for it happening on each chance. 878 million to 450k. Or about 900 mil / 450k = 2,000 to one. Not as exciting.

You can get it to look even less exciting if you consider that it doesn't even have to be a royal flush vs quads, it can be anything that is perceived as remarkable occurrence by someone who would post it on the forum.

Happens all the time, I dealt this very hand 3 weeks ago, only a £150 buy in tournament, so painful though, it's the dealers fault of course.. SIGH

Of course a question you might like to ask yourself is how many hands have been 'televised'.
Anyone like to take a guess at this?
Makes me think hands are rigged for action or TV drama.

Anyway a bit of maths, assuming 1 hand per minute (fast for live poker?).
It would take, 140 years of 24 hour poker, I made it 1,400 years if it per 10 player table, so I divided by 10, Im not sure if that is correct.

But anyway do you think there are 140 years worth of poker recorded for TV programs?

I would doubt it personally.

However having said that, I will 'bet against myself' and say that is just one of many 'freak' hands which could happen so maybe it is legit?