QQ verses AKo

[Able2call]

Why is it that AKo is 43% and QQ is only 56%. It seems that a 6 outer should not hit that much. I don't understand? Straights are limited by 2 Q's gone.
thanks

[Fenris78]

Well, AKo has "only" 6 outs that is true. I guess you are used to the calculation of outs and odds on flop and turn. When you get to see only 1 more card a 6 outer will not hit all too often that is true. But when you have 5 cards to hit a 6 outer it will happen quite often.

A simple calculation will show you this:

You have AKo, your opponent has QQ. That leaves 48 cards in the deck, 6 of which are As and Ks. Now you flip over the top card and see if it is and A or K. The probability that the top card is an A or K is obviously 6/48 or 1/8 or 12,5%.

Now repeat this but flip over the top five cards of the deck. You now have five chances to hit an A or K! What is the probability that at least one of those cards will be an A or K (we will ignore the fact that an Q could come or a AKo could make a straight for now)? The rough estimate would be 5 times 12,5% equal 62,5% which is close but not exact because you have to use inverse probabilites.

Thus it is equal to the probabilty of 1 minus the probabilty that no A or K will hit within 5 cards or:

1 - (42/48 * 41/47 * 40/46 * 39/45 * 38/44 * 37/43) = 57,3%

That means that the probability of an A or a K hitting in 5 cards is in fact higher than 50%! Look at it like this: the QQ effectively has to "dodge" an A or K 5 times in a row, which is quite often. Now luckily in 20% of the cases where an A or K comes (i.e. 11% of the time), there will also come a Q that will save the Q. Substract that from the times an A or K hits and you will see that the AKo will hit an A or K but no Q 45,9%.

Of course this calculation does not include the times where QQ or AK makes a straight (in fact the QQ can make more straights than the AKo as AK misses already 2 key cards that it needs to make a a straight thus improving QQs winning chance even more) or a flush (wether the QQ are the same suit as the A or K will influence the winning percentages a bit), but you see that with this simple calculation we are already very close to the correct percentages.

Hope this helps

[Bugsbunny]

Monte carlo simulation results from Poker Calculator 1.1.4.1
Texas Hold'em, 100000 combinations tested.

Hand 1:
Range of hands: AKs, AKo

Hand 2:
Range of hands: QQ

[Able2call]

Thanks guys.
it's the 5X drawing at an A or K as apposed to hitting 2 pair for a boat on just the river (1X).