What’s the best way to double your money? In a nutshell.

Further and deeper exploration of paradoxes and challenges of intuition and logic can be found in my recently published book, Probability, Choice and Reason.

John needs £216 to pay off an urgent debt, but has only £108 available. This is unacceptable to the lender and as good as nothing. He decides to try to win the money at the roulette wheel.

So what is his best strategy? The answer might be a little surprising. He should, in fact, put the whole lot on one spin of the wheel. Yes, that’s right. In unfavourable games (house edge against you) bold play is best, timid play is worst. Always place the fewest bets you need to reach your target.

Take the case, for example of a single-zero roulette wheel. So there are 36 slots and the zero and the payout to a winning bet is 35/1, while the chance of winning is 1 in 37 (so the payout should be at odds of 36/1). The way to look at it is that the house edge is equal to the proportion of times the ball lands in the zero slot, which is 1/37 or 2.7 per cent.  This edge in favour of the house is the same whatever individual bet we make.

So let’s see what happens when John goes for the ‘bold’ play and stakes the entire £108 on Red. In this case, 18 times out of 37 (statistically speaking), or 48.6 per cent of the time, John can cash his chips immediately for £216. Of course, he is only doing this once, so this 48.6 per cent should be interpreted as the probability that he will win the £216.

An alternative ‘timid’ strategy is to divide his money into 18 equal piles of £6, and be prepared to make successive bets on a single number until he either runs out of cash or one bet (at 35 to 1) yields him £210 plus his stake = £216.

To calculate the odds of success using this timid strategy, first calculate the chance that all the bets lose. So any single bet loses with a probability of 36 in 37. So the chance that all 18 bets lose = (36/37)18 = 0.61. Therefore, the probability that at least one bet wins = 1- 0.61 = 0.39. The chance that he will achieve his target has been reduced, therefore, from 48.6 per cent to 39 per cent by substituting the timid strategy for the bold play.

There are many alternative staking strategies that might put John over the top, but none of them can make it more probable that he will achieve his target than the boldest play of them all – the full amount on one spin of the wheel.

Exercise

You need £432 to pay off an urgent debt, but has only a bank of £216 available. This is unacceptable to the lender and as good as nothing. You decide to try to win the money at the roulette wheel.

What is the probability that you will win the target sum if you place all your bank on one spin of the wheel?

What is the probability that you will win the target sum if you divide up your bank and place £36 each on six spins of the wheel?

References and Links

StackExchange. How to win at roulette? https://math.stackexchange.com/questions/98981/how-to-win-at-roulette

Dubins, L.E. and Savage, L.J. (1960). Optimal Gambling Systems. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC223086/pdf/pnas00211-0067.pdf