Question about "average winning hand" - a mathematician said...

Jiggymike · #1 11-30-2006, 08:13 PM

Sorry if this is a stupid or ridiculous post but I was just thinking about something a friend of mine brought up. We were playing poker and I raised the first hand...everyone folded so I decided to show my AQo just for s+++s and giggles. Anyhow, he claimed he had a friend who is a mathematician as well as a good poker player (I cannot confirm the second part, he is afraid to invite him because the math guy apparantly wins a lot of money in poker online...luckily they never bother to inquire as to why I always beat them or have cards that say Full Tilt poker.com on them). The player remarked that his friend told him that after some analyses he found that AQ is your "average winning hand" in texas hold em. However, my friend wasn't sure what that meant and neither was I. Does that mean if you dealt out all the hands each and every time AQ would be your average break even hand against a full table? Anyone ever hear something in this vein? I know that Phil Gordon in LGB says that 2 pair is your average winning hand on a board.

alphatmw · #2 11-30-2006, 08:27 PM

is that supposed to mean that BB/100 with AQ = overall BB/100?

PantsOnFire · #3 11-30-2006, 10:12 PM

I would think it is impossible to figure out an "average winning hand". It would depend on the tightness of the game.

The only average hand you could figure out for sure is one where everybody played to the river and showed. Ask your math friend to figure that one out.

Jiggymike · #4 11-30-2006, 10:24 PM

[ QUOTE ]
I would think it is impossible to figure out an "average winning hand". It would depend on the tightness of the game.

[/ QUOTE ]

Yeh I told him that it made no sense but I was just wondering what other people thought of it.

Scary_Tiger · #5 11-30-2006, 10:37 PM

Makes no sense.

AaronBrown · #6 11-30-2006, 10:53 PM

There are several different things he might mean by that.

Rank the starting hands based on their chance of winning in a 10-player game (the number of players doesn't matter too much, except for some close calls). 1 to 6 are AA, 7 to 12 are KK, down to 1,315 to 1,326 for 72o. If you deal ten starting hands, on average the best one will be rank 121. That corresponds to K9s, but AQo is not too far away at ranks 89 to 100. With 13 players, it would be the average best starting hand. Given that your friend's friend might have defined things a little differently, this might be what he meant.

In a similar vein, he could mean that if you rank starting hands in poker terms, AQ is beaten by 94 of the 1,326 possible hands (13*6 = 78 pairs, plus 16 AK's) and tied by 16, making 110. In this case, with 11 players, AQ is the average best starting hand. With 10 players, it's AJ.

Another possibility is you deal 10 hands all the way to showdown, take the winning hand, and record the rank of its starting cards. This gives an average winning hand of A7o, which is only a slightly above average hand (in a 10 player game with everyone staying in to showdown, it wins 11.71% of the time, which is less often than 40% of the starting hands but more often than 59%). AQo is a much better starting hand, winning 16.78% of the time, more than 92% of the starting hands, and incidentally, dominating A7o.

None of this has much to do with real poker. You could instead rank hands by how much they actually win in play. AQ is well above average in this respect, it's the 12th best hand in 10-player tables, with an average profit of 0.26 BB. A3s is the worst hand with positive profit (0.001 BB) while T9s is the best hand with negative profit (-0.002 BB). But because of the rake, K8s is actually the average hand.

alanbrown · #7 11-30-2006, 11:27 PM

To go a little further on what Aaron is saying, let's say you're playing a game where instead of getting cards you get a real number between 0 and 1. If there are n players the average lowest number will be 1/n+1. And the highest number will be (on average) 1 - 1/n+1.

This means that if you're dealing out 10 starting hands, on average the best one will be in the 89th percentile.

That will be the 145th best hand. My hand list puts that at K9s (I obviously don't use the same list as Aaron as he had that hand at 121).

jjflash · #8 12-04-2006, 11:10 PM

Yeah and I once had a player tell me that J10s was the most winning hand in poker. She is always welcome at my game.
Take every brilliant thought that poker players have with a grain of salt because there is one math truth that reigns above all others. There are more losing players than winning players. Odds are you are talking to the loser.

roueful · #9 12-06-2006, 12:17 AM

Obviously we're guessing at this mathematician's intention, but I think he's talking about the set of hands that win an actual pot, and applying some kind of criteria amongst these winners to determine the 'average' one.

StewHunger · #10 12-13-2006, 03:56 AM

Where does one find the listing order of all hand rankings that you reference including the number of hands for that particular combo? For example, you say there are 1326 hands, the first 6 AA, next 6 KK, then 16 AK? or how is that broken down w/ suited? I guess what I'm getting at is I'd like to have an idea out of all possible preflop hands what represents top 5%,10,15, etc. I find this type of theory stuff very interesting so any tips on how you could figure this out on your own would be a bonus instead of just looking at a table.