PROBABILITY OF WINNING AT ROULETTE

# By Henry Tamburin

A reader wants to know: Since there are 18 red numbers on the roulette wheel and only one number 7, aren’t my chances of winning a lot better if I bet on red rather than betting straight up on a number?

Yes, you’ve got a lot more chances of winning on any spin if you bet on red rather than straight up on any number. The reason, of course, is because the probability of a red number hitting is greater than the probability of any one single number hitting on any spin.

Hitting red on any spin – probability is 18/38 or 47.4%

Hitting any number straight up on any spin – probability is 1/38 or 2.6%.

So, you’ve got almost a 20 fold better probability of hitting red than any single number. But does that mean betting on red is a better bet than say a bet straight up on number 7? No, it isn’t and here’s why.

What really matters when you play roulette is not how many times your number or color hits but how much money do you stand to win or lose. And that’s quite easy to calculate for any roulette bet.

Let’s take a bet on red. You can calculate the expected value of a \$1 bet on red by multiplying the probability that the bet wins (47.4% above) by the payoff and then adding to it the probability that the bet will lose times the corresponding amount lost. Sounds complicated but for a bet on red here’s what the equation looks like.

EV = probability bet wins x payoff + probability bet loses x amount lost.

We already know that the probability that red wins is 18/38 or 47.4%.

If the casino wins the payoff is \$1.

The probability that the bet loses is 20/38 or 52.6%.

The amount lost is \$1.

Now plug in the numbers in the above equation and you arrive at an expected value of

-\$0.0526.

Notice the expected value is negative. This means that you stand to lose money when you make a bet on red (about a nickel per dollar wagered). Why is that? Simply because there are 18 red numbers that will cause you to win and 18 black numbers plus the green 0 and double 00 (total of 20 numbers) that will cause you to lose. The two losing green numbers makes it less than a 50-50 proposition. In fact you’ve got only a 47.4% probability of winning on red and 52.6% of losing.

Now what about a bet straight up on number 7 where you get a paid substantially more - \$35 rather than \$1 for every dollar wagered.

If you run through the same equation for EV here’s what you get:

EV = probability 7 wins x payoff + probability 7 doesn’t win x loss

EV = 1/38 x \$35 + 37/38 x \$1

EV = -\$0.0526

Well, isn’t that interesting – you arrive at the same negative EV as a bet on red. Actually, if you compute the EV for every bet on the roulette table by the above equation you will

arrive at the identical negative EV of -\$0.0526 (with one exception – a bet on the five numbers 0, 00, 1, 2, 3 has a –\$0.0789).

The bottom line is that even though the probability of winning various bets in roulette is different the expected value is the same.

But even though the EV is the same for every roulette bet, the volatility isn’t. The latter is a measure of the fluctuation you can expect from an expected result. It turns out that bets that have a fairly large payoff but a low probability of winning have a high volatility. This means you stand a good chance of winning more or less than the expected EV of

\$-0.0526.

On the other hand, a bet on red has a low volatility because the bet has a fairly high probability of winning and the most you could win or lose is a dollar. The bottom line is that if you want your bankroll to last as long as possible it’s best to bet on red (low volatility). If instead you want to go for a big win at the risk of going broke sooner than bet straight up on a number (high volatility).