Explanation of the martingale fallacy

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The martingale fallacy is most commonly encountered using the even money bets available in Roulette. Here we will examine it in the context of the red/black roulette bet. If you need an explanation of the rules of roulette download the question set available here.

The thoroughly ingenuous strategy

Here is one round of the strategy.

On the first spin, bet $1 on either red or black. If your colour comes up you receive back your $1 stake and $1 profit and the round is completed. That is, you quit with a $1 profit. If your colour doesn't come up you lose the $1 stake, so your loss for the round so far is $1. Continue as below.
On the second spin, bet $2 on either red or black. If your colour comes up you receive back your $2 stake and $2 profit. Combining this $2 profit with the $1 loss from the previous step gives a $1 profit and the round is completed. That is, you quit with a $1 profit. If your colour doesn't come up you lose the $2 stake, so your total loss for the round so far is $3. Continue as below.
On the third spin, bet $4 on either red or black. If your colour comes up you receive back your $4 and $4 profit. Combining this $4 profit with the $3 loss from the previous step gives a $1 profit and the round is completed. That is, you quit with a $1 profit. If your colour doesn't come up you lose the $4 stake, so your total loss for the round so far is $7. Continue as below.
Continue for as many spins as required to produce a $1 profit for the round.

Once you have your $1 profit, you regard the round as over. You then start a new round, beginning again with a $1 bet. Continue for as many rounds as desired, with each round giving a $1 profit.

The difficulty is that within each round, the required bet doubles with each loss, so eventually you reach a point where either:

you don't have enough money to make another bet,
you come to your senses and realise it would be stupid to make such a large bet, or
the bet that you need to place to continue the strategy exceeds the maximum bet the casino will accept.

The marginally less ingenuous strategy

A slightly less misguided version of the strategy recognises there is a maximum number of spins that can be carried out. This number varies depending on where you read the strategy, usually falling in the range of 7 to 10. For our purposes I'll use 9 spins. With this maximum number, someone trying to con punters into using the stategy would have the bullet points given above ending as follows.

In the unlikely event that you ever reach the ninth spin, bet $256 on either red or black. If your colour comes up you receive back your $256 stake and $256 profit. Combining this $256 profit with the $255 loss from the previous step gives a $1 profit and the round is completed. That is, you quit with a $1 profit. If your colour doesn't come up you lose the $256 stake, so your total loss for the round is $511. But don't worry; the chances of that ever happening are vanishingly small.

Rubbish! The chances of that ever happening are large enough to make the game profitable for the casino and unprofitable for the person using the martingale strategy.

In a legal Australian casino the roulette wheel has 37 slots numbered 0 to 36. 18 numbers are red, 18 are black and the zero is green. Under this strategy the chance of losing 9 spins in a row is (19/37)⁹, or about 0.248%. That is, on about 0.248% of the rounds you play you will lose $511 and on the other 99.752% of the rounds you gain $1. On average your profit per round is about

99.752% × $1 + 0.248% × (-$511) = -$0.271

That is, on average you can expect to lose about 27 cents per round. This number scales with the size of the initial bet. That is, if you decide that winning only $1 on the successful rounds isn't enough and instead decide to start with an initial bet of $10, doubling after each losing spin to reach $2,560 on the 9th spin, then your expected loss is about $2.71 per round.

If you live in a country where the roulette wheels have 38 slots, with both a zero slot and a double zero slot coloured green, then the odds are worse. The chance of losing 9 spins in a row rises to (20/38)⁹, or about 0.310%, and on average you can expect your loss on each round to be about 58.7% of the size of the initial bet of that round.

The compact version

While losing 9 spins in a row happens often enough to ensure the above strategy is unprofitable, it happens rarely enough that it takes a lot of simulations to clearly demonstrate the scale of the loss. Hence the simulation page also lets you choose to run a simulation using a maximum of 3 spins.

The chance of losing 3 spins in a row is (19/37)³, or about 13.541%. That is, on about 13.541% of the rounds you play you will lose $7 and on the other 86.459% of the rounds you gain $1. On average your profit per round is about

86.459% × $1 + 13.541% × (-$7) = -$0.083

That is, on average you can expect to lose about 8 cents per round.

Loss rate

As explained elsewhere on this site, the payout odds for all the standard roulette bets are fixed at the level that gives the casino an expected take of 1/37 (about 2.7%) of the stake. Hence the simulation results should also show that in the long run the casino returns only about 97.3% of the amount bet. The 9-spin version of the simulation gives a greater loss per round than the 3-spin version (27 cents versus 8 cents) since on average the 9 spin version involves making larger bets. Under the 3-spin version gamblers can bet and lose at most $7 per round, where they can bet and lose $511 on the 9-spin version.

(Again, the above numbers are for a 37 slot wheel. On a 38 slot wheel on average the casino takes 2/38 (about 5.3%) of the amount bet.)

Run the simulation.

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