Friday, October 30, 2009

Amazing? Not really.

At Binion's tonight I was involved in a hand that got checked down on all three betting rounds after the flop. I won it with the low end of a straight. I didn't bet because by the river there were four hearts on the board, I didn't have one, and I was out of position against two other opponents.

When the hands were revealed, a player not involved said, "It's amazing that nobody had a heart!"

Apparently his definition of "amazing" and mine vary by a large degree.

Let's ignore the self-selection that usually occurs with poker hands by sequential betting (i.e., when the flop contains two hearts, players with one or two hearts as hole cards are more likely to continue playing than those with none), because here there was no such filtering action. What we are left with is the question of how likely it is that there is at least one heart to be found among six random cards (the hole cards of three players), given that four hearts are seen among the community cards. I will assume that before the flop, nobody is more selective for playing hearts than for any other suit.

We have 47 cards not on the board, which must include 9 hearts. So any individual card chosen at random has a 9/47 chance of being a heart, or 0.191. We use that as the value of p in a binomial calculation. My favorite binomial calculating tool is here. Using it, I learn that the probability of there being exactly one heart in a randomly selected group of six cards, under the circumstances described, is 0.40, or 40%. The probably of there being exactly two hearts among the six cards is 23%. Three hearts is 7%. Four hearts is 1%. Five and six hearts are vanishingly rare, accounting for only about 0.1% combined, so we can ignore those conditions.

In other words, the combined probability of there being at least one heart among the six down cards held by three players, when there are four hearts on the board, is about 72%. That means that about 28% of the time, none of the three players will have a heart. We experienced a condition that will occur 28% of the time--more than one time out of four. Not exactly a rarity. And this doofus considers that to be "amazing"?

Even if there were four players, and thus eight cards to account for, the probability that nobody has a heart is about 18%--again, not exactly a finding incredible enough to write home about. With five players, it's 12%; with six players 8%, with seven players 5%; with eight players 3.4%; with nine players 2.2%; and with ten players 1.4%. Those last two are about what it would take for me to be impressed that something truly out of the ordinary had occurred. But I still wouldn't call it anywhere near "amazing"--just kind of unusual.

I guess some people are a whole lot more easily impressed by minor coincidences than I am.


Michael said...

Semantics of course, but definitely up to the grump standards.

Anonymous said...

Amazing you would write about it.

The Vegas Flea said...

A 72% chance is pretty good in the gambling world. So, roughly 3/4 of the time someone is going to be holding a heart.

"Amazing" might be strong word to describe no one holding a heart, but with a 3/4 chance, a "Hmmmm, no hearts" might be more appropriate or perhaps a bland, "Wow, no heart?"

I get the feeling the choice of words here and your opponent's personality is the problem, not necessarily any poker probability.

Grump in true form. Well done.

THETA Poker said...

Another way to look at this is, "what are the odds of each hole card *not* being a heart?" The simple math is 38/47*37/46*36/45*35/44*34/43*33/42, which is 25.7%, a little lower than your estimate because your calculation simplifies to (38/47)^6, which overestimates all but the first term.

BTW, I apologize if it sounds like I'm picking on you. I'm really only one step ahead of you on this, but by going through this exercise, I'm trying to understand it all better. I just want to make my tiny contribution to your excellent blog.

Conan776 said...

What are the odds that no one takes a stab at the pot on the turn or river, though? I guess you don't stab at the river since you had a made hand, and maybe your opponents didn't either because they sort of had hands too?