The Gambler's Fallacy

And getting the analysis wrong.

I recently read (somewhere) which used this example:  Suppose you are playing blackjack, and you have lost 20 consecutive hands.  Are you more or less likely to win the 21st hand?  Whoever wrote that post (and I can't remember who, or where, and I haven't been able to find it) argued that your odds of winning the next hand are unchanged.

Well, maybe not.  Consider a coin flip (of a fair coin).  The odds of a head on any flip are 50%--because the flips are independent of each other.  Or suppose you are doing the other old standby--picking a marble from an urn, and you are (reliably) told that of the 1,000 marbles in the urn, 500 are red and 500 are white.  You pluck out a red marble, than toss it back in, and the urns is shaken to re-mix it.  Your picks are, again, independent of each other.

Now, suppose you are playing blackjack at a Las Vegas casino.  According to this source, the most common games use 6 or 8 decks at a time.  But the play is without replacement of the cards used in the current hand.  It's as if you drew a red marble from the urn and, instead of replacing it, you threw it away.  The odds of winning in a blackjack hand are, in fact, dependent  on how the play has gone so far from those 6 or 8 decks.  It is precisely the fact that blackjack hands are not independent that makes card-counting a viable strategy (and why casinos try to identify and throw card-counters out).

Any card game in which play is without replacement is a game in which your odds can change as play progresses.  You just have to know what sort of game it is.  And be willing to do the work to figure out how the odds have changed.  And counting a 8-deck shoe in a casino is hard, even if you work very hard at it.