Mathematical Prospect of Casinos
The essence of understanding mathematical expectations (also known as player expectation, expected value) is extremely simple. Strictly speaking, it is the amount of money you can win or lose over an extended period of time if you place the same bet.
You may want to calculate the value of mathematical expectation using the following formula:
МО = (number of positive outcomes [winner] / number of possible outcomes) * amount of wins + (number of negative outcomes [losses] / number of possible outcomes) * bet amount. Many of you will see this as a Chinese phrase, but it’s actually pretty simple.
You bet $ 1 that the first card will be a trophy. According to probability theory, the positive result (the trophy comes and you win and you win + 1 $) happens with the probability ¼, the negative result (you get another symbol card and you lose $ 1) happens with the probability.
Let’s calculate the mathematical expectation using the formula above:
МО = 1/4 * (1 $) + 3/4 * (-1 $) = – ½ $
So, over a long period of time, your loss will be 50 cents per dollar, so, according to math, you lose three times in 4 spins, you earn $ 1 each (you lose $ 3) and $ 1 at a time.
Mathematical Expectation in Roulette
Let’s calculate the mathematical expectation of roulette (American version with two zero sectors) when the $ 1 color (black): 1/18 * (+ 1 $) + 20/38 * (-1 $) bet. = -2 / 38 = -0.0526 (or -5.26%).
As you probably noticed, the value of mathematical expectation in both examples has a “-” (minus) value, which is possible with most casino bets. Negative mathematical expectation means the longer the game, the greater the probability of loss.
The casino advantage (House Edge) [casino percentage] is the value opposite to the mathematical expectation of the players; is the casino advantage (percentage) over the player. Casino advantage in European roulette 1 – 36/37 = 2.7%, American – 1 – 36/38 = 5.26% (thanks to two zero sectors). So that means; If you bet $ 1000, the odds of losing $ 27 (in European roulette) and $ 54 (in American roulette) are extremely high. The casino advantage is less in table games (Baccarat, Blackjack, or Craps).
Take American roulette again, which has 36 numbers and 2 zero sectors. Suppose we bet on a number. In this case, the probability of winning is between 1 and 36:
- The probability of winning: 1/38 or 2.63%;
- Possible win (stake percentage): 1/38 * 36 * 100 = 94.74%;
- Casino percentage: 100 – 94.7 = 5.26%;
- Mathematical expectation: (1/38) * 36 (+1) + (37/38) * (-1) = -0,0263.
- So, you can get 2.63 cents for every dollar bet in the casino. In other words, the mathematical expectation in American roulette is 2.6% from each of your bets.