Pocket Pairs Tripping UP

Last night in my weekly home game I had two people tell me that pocket 2's will trip up more often than pocket A's in Texas Holdem. They aregued this point for about an hour while i tried to teach them a little statistics. I guess it didn't work because they went home still arguing 2's trip up more than A's. I guess they think that the laws of mathematics apply in all of the world but not in cards or poker. I'm a young college kid and don't have much money but I offered to lay $1,000 on the table against one of them and they would have to run out the cards 10,000 times that there would be no statistical significance in the number of times that each will trip up. It will be roughly 50%-50%. Just wondering if anyone has a way to help me explain this to them. But until then I will put my pocket A's against their 2's all day long...lol.

[Tom1975]

You're absolutely right in a theoretical sense - if you gave one player pocket deuces and one player Aces and dealt 10,000 random boards, there would be no statistical difference.

Your friends may be right in a pratical sense, however. In a real poker game people are more likely to play hands with Aces in them than with deuces, which leaves fewer Aces than deuces in the deck on average. If you were to look a very large sample of hand histories I imagine you would actually see deuces making slightly more trips than Aces.

[MikeSmith]

Each is just as likely, regardless of whether aces get played or not. Thats a really off statement that aces get played more as a good argument. People get dealt a 2 just often as they are dealt an A. Who cares if people play aces more often, im gonna go throw up.

[Tom1975]

You misunderstood my post. I wasn't talking about people playing pocket Aces versus pocket deuces more often, I was talking about starting hands in general and how that affects the composition of the remaning deck.

Look at it this way. If you dealt out ten random hands and pretended that there were ten players playing them all the way to the river, then Aces would be as likely as deuces to trip up.

However, in a real poker game people do not play random hands. If you have Aces or deuces, in order to make trips, you need one or more players to stay in the hand so you can see a flop and have a chance to make trips. Since people are more likely to play a hand with an Ace in it than with a deuce, the very fact one or more people have decided to play makes it more likely that there are more deuces in the remaining deck than Aces.

Let's take an extreme example. Suppose you are playing heads up against an opponent who will only play his hand if it contains an Ace. In this scenario, every time you play pocket deuces and he calls, you know he has at least one Ace. Therefore, it is very likely that the two remaining deuces are still in the deck, except for the rare case where he has A2. Now let's look at pocket Aces. The only way you can make trips is if he plays his hand. If he plays his hand, it means he has at least one Ace, so there can be at most one Ace left in the deck, severly cutting you chances of making trips. If he has pocket Aces himself, you have no chance. Therefore in this scenario, you would make trip deuces about twice as often as trip Aces, since your odds of actually seeing a flop are lower when you have Aces.

it doesn't matter if the players in the game play hands with A's in them or 2's. it is just as easy for a 2 to be delt and folded as it is for an Ace to be dealt and played by someone. so therefore, there is no statistical difference because regardless if an ace is played or a 2 is dealt and folded it is not in the deck anymore and there is an equal chance of either card being delt from the deck when dealing players pocket cards

that is exactly true, but the two people arguing were arguing that if you have the two pocket pairs head up, 2's will trip more than A's. If they cannot understand the simple cocept here that it is 50-50 that each can trip up then i will not attempt to explain what u just posted although it is correct.

[Tom1975]

[ QUOTE ]
it is just as easy for a 2 to be delt and folded as it is for an Ace to be dealt and played by someone.

[/ QUOTE ]

This is true. However, if someone is dealt a 2 and folds pre-flop, and leaves you as the only remaning player, you have zero chance of making trips, since there won't be a flop. In order to make trips, there has to be a flop, which requiers that one or more players has been dealt a hand they deem playable, which means it is more likely they have an Ace than a deuce. See my responce to MikeSmith's post for a more detailed explanation.

people playing aces won't affect the deck...

here's what the point maybe should have been.

if you have a pair of 2's and there are 6-7 callers, then good chance all three 2's are still in the deck i.e. people would have folded Q2, J2 and even K2, A2.. but people won't generally fold aces in low limit, so if there are alot of callers (perfect odds for hitting 3 of a kind and winning huge), then it will be better to have 2's than A's for purposes of hitting 3 of a kind.

but without knowing the number of callers, no reason to think either is more or less likely.

[KenProspero]

The odds of a two showing up on the flop (or turn or river) when one player was dealt pocket twos is exactly the same as the odds of an Ace showing up on the flop (or turn or river) when one player was dealt pocket Aces.

However, to make a semantic (though not probababilistic) argument, the chances that the player with the pocket 2s will fold before the third 2 shows up are much greater than the chance that the the player with Pocket Aces will fold before the third Ace shows up.

So, here's the question -- if someone folds pocket twos, and the flop contains a 2, have they tripped up. I'd argue not. If you agree with that argument, then aces trip up more than twos.

I sort of doubt that your friends were making this argument, and think they were probably talking mathmatically, which is nonsense.