How To Calculate Player Advantage In European Roulette

by Mark on July 21, 2009

In European roulette, the formula to calculate player advantage percentages for any bet can be calculated by:

\frac{\text{number of favourable results } \times \text{ payoff for favourable result}}{\text{total number of possible results}} + \frac{\text{number of unfavourable results } \times \text{ the bet amount}}{\text{total number of possible results}}

A Simple Example: Straight Up Bets

For example, if a player wins straight up (ie, a single number) on a European roulette wheel (36 numbers plus a single zero), he will win $35 for a $1 stake. The formula gives us:

\frac{1 \times 35}{37} + \frac{36 \times -1}{37} = \frac{35 - 36}{37} = -2.70\%

There’s a few things to understand here. First, you’ll see the result is negative. This means that the advantage is to the house, not the player. Over time, players will lose an average of 2.70% of their stake, per bet, to the house. If the number was positive, then over time the house would lose that average of 2.70% of each stake back to the players.

Let’s highlight with an example. Assume the payoff was $36 for a straight up win, instead of $35. The formula would give us:

\frac{1 \times 36}{37} + \frac{36 \times -1}{37} = \frac{36 - 36}{37} = 0.00\%

There’s no advantage to either house or player in this case; over time, both the house and the player would not win or lose anything.

Let’s take it one step further. Assume the payoff was $37 for a straight up win. The formula is now:

\frac{1 \times 37}{37} + \frac{36 \times -1}{37} = \frac{37 - 36}{37} = 2.70\%

Over time, the player would average winnings of 2.7% per bet. If you can ever  find a casino that offers these odds, keep quiet about it and play there often!

Outer Bets

Now I’ll move to the very popular outer bets in roulette, which are, of course black vs red, odds vs evens, or low numbers 1-18 vs high numbers 19-36. Let’s say you bet $1 that the number will come up red; if it does, you win $1 (plus you get back your stake of $1). Using the formula:

\frac{18 \times 1}{37} + \frac{19 \times -1}{37} = \frac{18 - 19}{37} = -2.70\%

Here’s one of the things I like about roulette – you can take a straight up bet that seems risky on the surface of things, yet it has the same potential payout as the bet that seems least risky. I’ll get into this in another post.

Of course, the player advantage is the same for all outer bets.

Dozens Bet

Finally, let’s have a look at the dozens bet, where you wager on one of three groups – 1-12, 13-24 or 25-36. The payout for these dozens wagers is $2 for a $1 stake, or 2:1. The formula looks like this:

\frac{12 \times 2}{37} + \frac{25 \times -1}{37} = \frac{24 - 25}{37} = -2.70\%

Once again, it’s exactly the same player advantage as for outside bets or straight up bets.

A Concluding Thought

Get it? By now, you should have a good idea of how to apply the formula to calculate your advantage for any roulette bet. And when you do, you’ll see that all the simple (ie, non specialty) bets on a European roulette wheel offer the same player advantage  of -2.70%.

What this means is that it really doesn’t matter whether you keep your money on a single number in a straight up play for a big win, or you’re happy to have a small win on the outers every other spin or so. Your chances of winning are the same. So have fun and bet wisely!

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by Mark on July 8, 2009

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