Poker shared cards change odds

Question

I wrote a poker simulator for seven card texas hold 'em, and I found a counter-intuitive result:

If player A starts with a pair of aces against player B who has a random hand, I get a different %win if they share five cards on the board (as it works in Texas hold'em) than if they each have separately drawn hands of seven cards. In the standard Texas Hold'em procedure player A has an 85% chance of winning, but if I draw two seven card hands separately (instead of sharing five cards) then player A has a 77% chance of winning.

In both scenarios the cards are drawn from a deck of 50 cards (ie a normal deck excluding the two Ace's I've given player A at the beginning).

Between the two scenarios the probabilities I measure of player A and player B having a high card, a pair, two pairs etc are unchanged .. but somehow player A wins more often in one scenario than the other.

I've tested lots of times in different ways, and I'm convinced it's not due to a bug in my program. Can someone help me with an intuitive picture of what is going on?

Answer 1

I don't know if what your simulator is doing with tie percentages, but it's likely due to paired boards, 3 of a kind boards not helping the inferior starting hand if the board is shared. For example.

Your hand: AA

My random hand: JJ

Board runs out 22347, we both make 2 pair. In fact, any pair on the board or 3 of a kind 22237 doesn't help my hand, b/c I started the hand behind and we're sharing those cards. If I get to have my own board, then making a second pair or a full house will help me. In fact, sometimes I would be able to start with 2-7 and if my board was

44336, I would have made two pair and could beat your lone pair of aces, while if the board was shared, I'd have 2 pairs 4's and 3's, and you'd have Aces and fours and win the hand. That's likely a great percentage of the difference. Having your own board simply gives the inferior starting hand more chances to pull from behind. Which is exactly why more "bad beats" are dealt in stud games then flop games.

Answer 2

The 2 poker hands will not be independent in either case, but the level of dependence will be higher when the cards are shared. Winning percent will be highly influenced by the level of dependence.

If the shared cards include another ace, that helps you a lot more than your opponent. If the cards are not shared and your opponent gets an ace then that could help them, but will lower your chances of getting 3 or 4 of a kind. If the shared cards include KQJ10 then that gives you a straight, but will only help your opponent if they also have an ace.

Probabilities of flushes will also differ.