Community cards and hand strength

[senjitsu]

Given the same number of players, is the average hand strength likely to be the same for 8 random stud hands and 8 random holdem hands.

Put another way, does the community card aspect of holdem make big hands (like quads or royal flushes) less likely?

My sense of this is "yes", but recently i ended up in a discussion about it and couldn't figure out any mathematical way to express why. The other guy's arguemnt was that you make the best 5 card hand from 7 random cards in both games, so any given hand strength had to be equally likely.

[Siegmund]

The community card aspect makes the distribution "clumpy" - there are some boards were noone can possibly have a good hand, and others where bad beat jackpots are likely to happen.

But the overall average is the same. Law of total probability: E(X) = E(E(X|Y)) over a set of Y's that span the probability space. In English: asking "what's the average poker hand for 7 cards" is the same thing as asking "what's the average poker hand on this board" for every possible board and then averaging your answers.

[AaronBrown]

Although Siegmund's answer is correct, I think I understand your intuition.

Suppose you get sent to purgatory and have play poker until someone gets a royal flush (this is like that Twilight Zone where Hell is getting everything you want). You get to choose whether to play 7 card stud or Hold'em. Assuming for some reason that you want to get out early, you'd choose 7 card stud (unless you really like Hold'em a lot better).

The expected number of royal flushes after any number of deals are the same. But 1 time in 21 when there's a royal flush in Hold'em, there are 8 royal flushes because it's on the board. This wastes a lot of your expected value, because you can't get out of purgatory 8 times over.

After 10,000 hands, for example, you expect 0.1231 royal flushes. Since the odds of getting more than one in 7 card stud is small, the expected value is close to the probability. If you can only get 0 or 1, probability and expected value are the same thing. You have about an 11% chance of being out of purgatory before the 10,000th hand at 7 card stud.

In Hold'em, there is about 0.5% chance that you'll have 8 royal flushes, this consumes 4.2% of your expected value. So you only have about a 7% chance of getting out of purgatory after 10,000 hands.

Of course, if they don't allow Poker in heaven, you'd have to reverse the calculation.