*and*another ace flopping...

Here is a summary from Doyle on the probability (we will do a reality check on this first).

"In a 9 handed game when you have 2 kings... a player will have an ace about 80% of the time."

"... An Ace will flop about 18% of the time..."

So, first let us check the 80% statistic. There are 6 ways each player could hold AA, and 6 ways for AK (you have KK after all) and 12 ways for each of the other A-x cards, this adds up to each player having a probability of 138 / 1326 - 9.6% chance of holding an ace... (NOTE: If you are looking for how I calculated these numbers then check out this post: http://plan3tgongpoker.blogspot.com/2007/05/poker-flop-probability-continuation.html )

Well, thats not quite 80% but close enough - the chances of more than one player being dealt an ace are significantly reduced by any ace already being in the hand of another player. Conversely if 6 players are not dealt an ace then the chances of the remaining players getting one will go up (this is the 'Bunching Concept'). Also it may not be accurate from a statistical standpoint to simply sum the probabilities of each player getting an ace... after all each 9.6% is a separate calculation.

For now we will work with the 77% rough calculation.

The next point here is that not all aces... sure some players like A-4 offsuit, but most do not. So you raised with your KK before the flop - which of those aces are calling your raise?

We will try 2 different levels, A10 + and A7+ so that this concept can apply to a few different buy-ins...

A7+ Calling your raise - Probability = (78/1326)*8 = 47%

AT+ Calling your raise - Probability = (48/1326)*8 = 29%

Fine, but there are 2 things more to work out before these statistics become useful. Firstly the chance of an Ace coming down on the flop... secondly the proportion of A-X hands you opponent might hold in relation to the other hands he would have called with (pairs etc). Will look at the first one today and make a part #2 out of the 2nd (big enough for a whole post of its own!!).

There are 3 aces (mostly) left in the deck - and 50 unseen cards (you can only be sure of the two Kings in your hand). So the chance that each of the 3 flop cards is an Ace = 6%, actually if the first card is not an ace then the chances for the 2nd go up... the actual equation is

(3/50)+(3/49)+(3/48)

Thats around 19%, not too far from Doyles 20% really. Remember that in our scenario there are still 2 kings in the deck, thats just over 12% to flop one (or just over 8% if one of the flop cards is an ace).

All very nice, but how can this be used. Well it should emphasis the importance of getting money in pre-flop with Kings! Yes, there is a fair chance that at least one opponent holds and ace. Yes, an Ace could flop killing your hand (assuming one of the remaining Kings does not).

Of course there are more hands that an opponent could have called with than A-X, that will wait for part #2 (Sunday - going to do the weekly digest tomorrow).

GL at the tables, Mark

PS: Have been working hard on SNG planet this week and now have a Poker Tournament Beginners Area up and running, includes a tournament FAQ, Quick Start Strategy Guide and tournament selector matrix... + more to follow.

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