Investigating Roulette

In previous Newsletters, we’ve looked at the chances of winning prizes on the UK National Lottery, the “birthday conundrum” and the well-named Gambler’s Fallacy. I’m continually pleased at how many clients give me positive feedback about this column, so thank-you for your kind comments and long may it continue!

For this issue, I thought we’d put on our tuxedos, order a Martini, “shaken, not stirred” and investigate Roulette…….

Although I personally don’t play in casinos, Roulette is by far the most popular of casino games, taking about 60% of total stakes. For beginners, the standard UK roulette wheel has 37 numbers, 0-36. The number 0 is coloured green with the other numbers equally divided between red and black. Players buy an equivalent number of coloured chips and disputes don’t arise about who has won, as all players have different coloured chips. The maximum stake is usually 100 times the minimum, with different tables in the casino offering different stake levels.
We can assume that all 37 outcomes, when the wheel is spun, are equally likely to occur. Surprisingly, casinos have a powerful incentive to meet this ideal as their advantage is so small that any detectable bias, might tip the odds in favour of alert punters.

To find out the house advantage, we must consider what will happen on average.
For any bet, other than those that pay out even money, the average return is 36 units for every 37 staked. But players combine bets, miss turns, change the size of their stakes etc however, this average remains constant. 1 unit is lost for every 37 staked, which is an advantage to the casino of 2.7%. We can now plainly see why there are no clocks in casinos, intending to make gamblers play for longer. With an advantage like this, the longer casinos can get their punters to play for, the more money they will make. That’s the casino, not the punter!

With bets on red or black only (with no number chosen just the colour) these pay out even money. In this situation the casino advantage is half as much. This is because of the rule for these bets that when 0 is spun, the house takes half the stake, with the other half returned to the punter.

Consider Paul, who needs £216 to buy an airline ticket, to watch his team play in the European Cup Final tonight. He enters the casino with exactly half the amount needed (£108) but might just as well have nothing, as he can’t get his ticket, even if he had £215. He hasn’t got a credit card and none of his friends can lend it to him.

Can a visit to the casino help and what’s the best strategy to advise him with?

Faced with this sort of problem, we can offer some good initial advice:
“In unfavourable games, bold play is best, timid play is worst”. To simplify our situation, we’ll assume that even money bets also lose when the 0 is spun, so the casino advantage is the same regardless of which bet he makes.

The first bold play might be to put the whole £108 on red, then 18 times out of 37 (48.6%) he can cash his chips in and go to the game, otherwise all his money is lost.

There are other bold approaches he could employ but will it improve his chances?

He could divide his money into 18 piles of £6 and bet successively on a single number until, he’s either won or his cash has run out. Single numbers pay 35:1, so one win is enough. To calculate his chance of success, we first find the chance that all his bets will lose. Any bet loses 36 times in 37, so the chance he will lose is (36/37) raised to the power 18. This gives 0.61 or 61%, so his chance of at least one win is 39%. This is rather worse than one single bet on red.

An alternative for Paul to use the 35:1 single number betting is if he divides his £108 into several different amounts. If he uses just £4 on the first bet, a win will take him to his target. In fact, as long as he has at least £76, a winning bet of £4 stake is enough to win. With less than this, he should stake £5 and when he has less than £41, he should increase his bet to £6. In this way, he can plan to have a total of 22 bets if necessary. If any of his bets wins, he wins enough for the flight; if not he’s left with £1 and will have to watch his team on TV.

As we did above, we start with the chance they all lose, which is (36/37) to the power 22. This gives 54.7%, so he can expect to succeed only 45.3% of the time.

Clearly the bet of all his cash on one big red bet is still his best option.

The numbers I’ve used were to make the mathematics easier but it proves the initial advice we gave Paul was good. “Bold play in unfavourable games is best”.
Bold play here means making as few bets as possible. His overall expectation of buying his plane ticket is low but he still seeks his best chance.

What would you have done?

I’d probably have bought a few cans of beer and watched it in the comfort of my lounge…..

More probability adventures in the next Newsletter.

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