The roulette wheel of fortune

The game of roulette is a wheel of fortune, unfortunately not for you but rather for the casino. Why? Five words – the law of large numbers. The people running the casino know these five words and their implication. They know how this principle of probability will impact their bottom line. Not so some of the gamblers. In the long run, the casino will win 5.26 cents for each dollar wagered at the roulette table (i.e. the house has an edge of 5.26%). A player may feel lucky on a given night at the table. But if the player plays long enough, the casino wins. The results of individual bets made at the roulette table may be unpredictable. But the long run average of making thousands of bets at the roulette table is predictable and stable. This is the essense of the law of large numbers. In this post, I discuss the game of roulette and demonstrate how the house edge of 5.26% is derived.

The American roulette wheel has 38 congruent sectors numbered 0 and 00 and 1 through 36. The sectors 0 and 00 are green. Half of the sectors 1 – 36 are red and the other half black (see the following photos).

The payout of a bet on the roulette table is expressed as h to 1. This means that for a $1 bet, if the player wins the bet, the player keeps his own $1 and in addition gets $h from the casino. Otherwise, the player loses his $1 to the casino. If the payout is 1 to 1, then the bet is said to be an even bet.

In spinning the roulette wheel, the ball can land on any one of the 38 sectors at random. If the player places a bet on the red for example, then the odds against the player are expressed as 20 to 18 since there are 18 sectors in red.

If the player places a bet on a single number (say 15), then the odds against the player are 37 to 1. In general, for a given bet, the odds against the player are expressed as p to q where p + q = 38 and the number q is the number of sectors that correspond with the event that the player winning the bet. Stated as probability, the probability of the player winning a given bet is \displaystyle \frac{q}{38}.

There are many types of bets you can make at the roulette table. Let’s consider three of them – a bet on color (red or black), a bet on a single number, and a bet on 3 numbers (any row). The house edge for these three bets (and any other type of bets that you can make at the table) is always 5.26%.

For the bet on color (red or black), the payout is 1 to 1 and the odds against the player are 20 to 18. The probability that the player wins the bet is \displaystyle \frac{18}{38}. For a $1 bet, if the player wins, in addition to keeping his $1, he gets an additional $1 from the house.

For the bet on a single number, the payout is 35 to 1 and the odds against the player are 37 to 1. The probability that the player wins the bet is \displaystyle \frac{1}{38}. For a $1 bet, if the player wins, in addition to keeping his $1, he gets an additional $35 from the house. This sounds great. But the chance of losing is much higher (on average 37 wins for the casino for each win of the player).

For the bet on 3 numbers (any row), the payout is 11 to 1 and the odds against the player are 35 to 3. The probability that the player wins the bet is \displaystyle \frac{3}{38}. For a $1 bet, if the player wins, in addition to keeping his $1, he gets an additional $11 from the house. The ratio of the house winning bets to the gambler’s winning bets is 35 to 3.

How is the house edge calculated if the bet is on the color (red or black)? The odds against the player are 20 to 18. The payout is 1 to 1. In the long run, for each 38 bets of $1, the player on average wins 18 bets (gaining $18) and loses 20 bets (losing $20), with a net loss of $2. Thus for each $1 in wager, the average loss for the player is \frac{2}{38}=0.0526 ($0.0526 or 5.26 cents).

How about the house edge if the bet is on a single number? The odds against the player are 37 to 1. The payout is 35 to 1. In the long run, for each 38 bets of $1, the player on average wins 1 bet (gaining $35) and loses 37 bets (losing $37), with a net loss of $2. Thus for each $1 in wager, the average loss for the player is \frac{2}{38}=0.0526 ($0.0526 or 5.26 cents).

Now let’s look at the bet on any 3 numbers (any row). The odds against the player are 35 to 3. The payout is 11 to 1. In the long run, for each 38 bets of $1, the player on average wins 3 bets (gaining 3 times 11 = $33) and loses 35 bets (losing $35), with a net loss of $2. Thus for each $1 in wager, the average loss for the player is \frac{2}{38}=0.0526 ($0.0526 or 5.26 cents).

The following matrix lists out some of the common bets placed In American roulette.

\displaystyle \begin{pmatrix} \text{Type of Bet}&\text{Payout}&\text{Odds Against Player} \\{\text{ }}&\text{ }&\text{ } \\\text{Color (Red or Black)}&\text{1 to 1}&\text{20 to 18} \\\text{Parity (Even or Odd)}&\text{1 to 1}&\text{20 to 18} \\\text{18 numbers (1-18,19-36)}&\text{1 to 1}&\text{20 to 18} \\\text{12 numbers (columns or dozens)}&\text{2 to 1}&\text{26 to 12} \\\text{6 numbers(any 2 rows)}&\text{5 to 1}&\text{32 to 6} \\\text{4 numbers (any 4 number seq)}&\text{8 to 1}&\text{34 to 4} \\\text{3 numbers (any row)}&\text{11 to 1}&\text{35 to 3} \\\text{2 numbers (adjacent)}&\text{17 to 1}&\text{36 to 2} \\\text{single number}&\text{35 to 1}&\text{37 to 1}\end{pmatrix}

The house edge (also called the expected value) is the expected percentage of the player’s money that is expected to go to the house. For any type of bets that can be made on the American roulette wheel, including the ones indicated in the above matrix, the house edge is always 5.26%.

When performing a random experiment (e.g. making bets at the roulette table), the individual observations are not predictable. However, the long run average of many independent observations is predictable and stable. This is called the law of large numbers. For example, there is no way one can predict whether a player will lose on any given bet at the roulette table. In any one bet, the player may win or lose. But in the long run, the player will lose (and the casino wins).

For anyone playing the game of roulette, the game stops when the player decides to stop or when his or her money runs out. The latter will inevitably happen in the long run (the law of large numbers). The game of roulette is a great game. Just know the odds.

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3 Responses to The roulette wheel of fortune

  1. Pingback: Celebrate the year of the rooster | All Math Considered

  2. Pingback: Playing the roulette wheel of (mis)fortune – Talking About Numbers

  3. Pingback: Law of large numbers – A World of Ideas

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