Smash Vegas

You’ve tossed a fair coin 9 times and it’s come up heads every time.  The odds of a fair coin coming up heads 10 times in a row are 0.5¹⁰ = 1/1024.  So if you were asked to bet on the outcome of the tenth toss the odds would favour you choosing tails, right?

Of course the answer is no.  No matter how many times you toss a fair coin it is still 50/50 that it will land heads or tails.

That’s because the coin has no memory and each throw is independent of the last.  Not recognizing this and thinking that a result is “due” to “balance out” the probabilities is the gambler’s fallacy.

It’s an easy mistake to make because we assume that short term trends should be representative of long term trends.  We know that in the long term that coin will come up heads about 50% of the time so we expect deviations from the average to even out.  We fail to understand that in the short term a run of coin tosses is unlikely to “look random” and clustering of results is the norm.

The key to avoiding this fallacy is recognizing when a game has memory and when each round is independent of the last.

Throwing a coin obviously has no memory.  While playing texas hold’em getting dealt pocket aces twice in a row does not change the odds of you being dealt them again in the next hand (provided the deck is shuffled between hands).

On the other hand since the deck in blackjack is not shuffled every hand, a high occurrence of low cards does mean that more high cards are “due”.  This is because the hands are not independent, and is the basis of card counting.

The gambler’s fallacy is at the root of many fraudulent betting systems, don’t get caught out by it.