What's the odds pocket Aces will hold up?

My brother was telling me about a hand he played last night in an online Hold-Em tourney. He got crap hands for the first 25 hands of the tourney and did nothing but fold, then, as Big Blind, got pocket Aces.

There were FOUR all-ins before he even got a chance ot bet.

He KNEW that meant that at least one ace was already out there, probably at least one other pocket pair, etc, but he called anyway, and lost to someone else's pair making a set.

His question is, what were the odds it would have held up?

He knows that heads-up, Aces are a big favorite, but against four other players?

I told him:
It doesn't matter how many players were dealt.
It doesn't matter how many players went all-in or when they went all-in.
It only matters how many players were still in at the showdown.

Am I right?

And what WERE the odds?

Thanks

In a previous WoO post, the Wizard but the odds at somewhere between 31.36% (10 players, no one folds) and 70% (estimate of real conditions):

Quote:

Wizard of Odds (https://wizardofodds.com/askthewizard/155 ) I've been a huge fan for many years (even before you got interested in poker and sports betting) and looked forward to every Ask The Wizard column. It's great to see you're doing them again! My question is this: at my local card room, they offer Aces Cracked, Win A Rack during certain hours. That is, if you have pocket Aces in one of their 3-6 or 4-8 Texas Hold 'Em games and you lose the pot, the casino will give you a rack of chips ($100). I'm trying to figure out how often a)I get pocket Aces b)how often they would lose if I played them aggressively as I'm supposed to and c)whether it's not better to just check all the way down and hope to lose, as $100 is usually better than what the pot would have been anyway. Any stats you may have at the ready would be wonderful and forever appreciated! Thanks again and keep up enlightening the masses! - Shane from Santa Rosa

Thanks for the kind words. The probability you will get pocket aces in any one hand is 6/1326, or once every 221 hands. According to my 10-player Texas Hold 'em section (/holdem/10players.html) the probability of winning with pocket aces is 31.36%, assuming all players stay in until the end. However that is a big if. If forced to make a guess I'd estimate the probability of winning with aces in a real 10-player game is about 70%. So the probability of getting pocket aces and then losing is 0.3*(1/221) = 0.1357%. So, at $100 per incident that is worth 13.57 cents per hand. Over ten people that costs the poker room $1.36 per hand on average, which cuts into the rake quite a bit. I tend to agree with your strategy of calling, which will keep more players in the hand, and increase your chance of losing.

I feel for your brother, but pre-flop, he had no choice but to call. It would have been different if he could have isolated some other players, but with the four all-ins, he was no worse than tied for the best hand at the time.

He was definitely an underdog, no matter what the other four hands were. He was, to simplify the calculation a bit, an 80% favorite against any one person, but since he had to beat all four, he was (.8)(4) to win the hand, or about 42%.

In a cash game, folding Aces preflop is a very dumb move. In a tourney, it would be similarly dumb, except in some situations where the holder of the Aces is near cashing, or already in the money with very little chance to move up. An extreme example would be five players left, four of them have $100,000 in chips and the fifth has $5,000. Before the fifth player acts, the first four go all in. The fifth player shouldn't even look at his hand, but even if he does, and sees pocket Aces, he should still throw his hand away.

Given that this was still early in the tournament, however, your friend had absolutely no choice but to call. He should have thought of this as an opportunity--getting 4-1 odds when he was only about a 4-to-3 underdog!

Quote: mkl654321

Given that this was still early in the tournament, however, your friend had absolutely no choice but to call. He should have thought of this as an opportunity--getting 4-1 odds when he was only about a 4-to-3 underdog!

Only an underdog against the entire field, right? But a favorite against any individual member of the field? To me, that distinction matters. For example, if the 5 players had win shares of 42%, 28%, 15%, 10%, 5%, then the guy with a 42% chance of prevailing should be viewed as the favorite.

lets do an exercise on worst case:

AA - 1.86% win, 43.32% tie
AA - 1.86% win, 43.32% tie
KQs - 14.75% win, 0.2% tie
99 - 20.9% win, 0.2% tie
33 - 17.3% win, 0.2% tie

assume everyone put in 1000 chips.
Equity is:

AA - 93 + 1083 = 1176
AA - 93 + 1083 = 1176
KQs - 737.5 + 10 = 747.5
99 - 1045 + 10 = 1065
33 - 865 + 10 = 875

even in this case, the AA hands and the 99 hands are "good" bets as they have a positive equity against the cost to play.
that's really where it stands, forget win%, think equity.

Quote: rdw4potus

Only an underdog against the entire field, right? But a favorite against any individual member of the field? To me, that distinction matters. For example, if the 5 players had win shares of 42%, 28%, 15%, 10%, 5%, then the guy with a 42% chance of prevailing should be viewed as the favorite.

Yes, he is the favorite, but his most likely outcome is that he will lose. Just like a horse that has the lowest price on him at, say 2-1. Or how the Yankees (or whomever) might be a favorite in any head-to-head matchup, but they are still a big underdog to win the World Series.