What is Gambler’s Fallacy?

Few fallacies are more persistent in gambling circles than the belief that the next toss (or spin, or draw) will somehow be influenced by the last one. Gamblers, and others, are led into this fallacy by confusing the odds against a whole sequence with the odds against any event in that sequence.

The odds against a tossed coin coming down heads five times in a row are easy to calculate. The answer is


1/2 x 1/2 x 1/2 x 1/2 x 1/2, or 1 in 32


If the first four tosses, despite the odds, come down heads, the chance of the fifth toss being heads is not 1 in 32, but 1 in 2, as it

was for each of the other tosses. The previous tosses do not affect the odds for the next one. In random or chance events, each go is separate from previous or future ones. Most casual gamblers, seeing four heads in a row, would bet on tails for the fifth toss because five in a row is unlikely. The professional gambler would probably bet on heads again, suspecting a crooked coin.


Red has come up 73 times out of the last 20.

That means we are due for a run of blacks. I am betting on black.

(If the table is honest, the odds on black remain, as before, the same as the odds on red.)

There is widespread belief in everyday life that luck will somehow even out. The phrase ‘third time lucky’ is indicative of a general feeling that after two failures the odds for success improve. Not so. If the events are genuinely random, there is no reason for supposing that two losses improve the chances of a win. If, as is more common, the results reflect on the character and competence of the performer, the two losses begin to establish a basis for judgement.


I’m backing Hillary Clinton on this one. She can’t be wrong all the time.

(Oh yes she can.)


One area where previous events do influence subsequent ones is in the draw of cards from a limited pack. Obviously, if one ace is drawn from a pack of 52 cards containing four aces, the chances of another ace being drawn are correspondingly reduced. Professional gamblers are very good at remembering which cards have been drawn already, and how this bears upon forthcoming draws. Still other gamblers are very good at making up from their sleeves what the laws of chance and probability have denied them from the deck.

Many so-called ‘systems’ of gambling are based on the gambler’s fallacy. If betting on a 1 in 2 chance, you double the stake after each loss, then when you do win you will recover your losses and make a modest gain. The trouble with this is that the rules of maximum stake, if not your own resources, will soon stop you doubling up. (Try the trick of doubling up ears of wheat on each square of a chessboard, and see how quickly you reach the world’s total harvest.) Furthermore, the odds are that the sequence which it takes to beat such a system will occur with a frequency sufficient to wipe out all of the winnings you made waiting for it. Only one rule is worth betting on: the house always wins.

You can use the gambler’s fallacy by appealing to a quite unfounded general belief that the universe is somehow fair.


My argument for avoiding the west of Scotland is that it has rained there on about half the summers this century. Since it was fine for the last two years, the odds are that it will rain this year.

(Things change, even in the west of Scotland.)


You may find the gambler’s fallacy particularly useful in persuading people to go along with you, despite a previous record which indicates that luck was not involved.


I propose this candidate for our new secretary. I know that the last three I chose were pretty useless, but that’s all the more reason to suppose I’ve had my share of the bad luck and will be right this time.

(This sounds like bad judgement disguised as bad luck. The odds are that the new choice will be both pretty and useless.)

The last four lawyers I had dealings with were all crooks. Surely this one must be better.

(No chance.)