Friday, April 2, 2010

On Statistical Anomalies

(the table lists Hand #, Table Name, My Hole cards, Winner, Pot)

Having nothing to do with linguistics, I challenge my fellow online poker player Nate Silver to walk through the probability that I would be dealt pocket 22, 33, 44 successively in NLHE. I have proof positive that it happened (see image above). And I note that the probability of being dealt any three pairs in a row should be the same as the probability of being dealt three consecutive pairs; it's us silly humans who care about the difference between 22 and KK, not the poker gods.


Anonymous said...

The probability of being dealt any three pairs in a row is not the same as being dealt three consecutive pairs in a row.

On any given hand, you have a 6% chance of being dealt a pair as your hole cards (first card at 100%, then the match of 3 out of the remaining 51 cards; the other players' cards are irrelevant as you have no information about them). 6% x 6% x 6% = 0.02% chance of drawing any three pair in a row. A small, but hardly earth-shattering chance of occurring.

But with three consecutive pair, the first hand narrows the range required for the next two, thus the odds are considerably smaller. The first hand is the same 6%, any pair will do (assuming you count AA-22 as consecutive). But for the second hand, you must draw one of four cards out of 52 (8%), then one of three cards out of 51 (6%), for a combined probability of 0.5%. You must do this again for the third hand, yielding a total odds of 6% x 0.5% x 0.5% = 0.00015%.

Note these are the odds for a given set of three hands. But given the number of poker hands played worldwide each day, it probably happens a few hundred times a day. And if you play enough, the odds that it will happen to you at some point in your gambling career will be rather high.

Chris said...

Anonymous, yep, you're right. I see your logic.

I guess I was thinking about the odds of being dealt three specific hands in a row. If I'm correct, the fact of their being consecutive in a number line are irrelevant to the odds that they'll be dealt on any given hand; it's just the fact of their being specified ahead of time (hence narrowing down the requirements as you say). Is that correct?

I think the point I wanted to make was that people have an illusion that poker hands that are related in a number line like 22, 33, 44 are somehow less probable to be dealt near each other in a sequence of hands than those that are not, when in fact, it is irrelevant. The number are just symbols printed on a card.

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