It isn’t exactly a secret that casino games are rigged against the player and in favor of the casino. Of course, casinos would not be massive cash cows bringing money to the tune of billions without an edge. But nevertheless, the ways casinos manipulate probability and math can show just how rigged they are.
Before delving into casino math, we have to start by talking about the concept of expected value. Expected value essentially consists of taking a weighted average of results by how frequently they occur. For example, if I told you you could play a game with me where there is a 1/3 chance I give you $3 and a 2/3 chance I give you $9, the expected number of dollars you receive would be ($3 * 1/3) + ($9 * 2/3) = $7. These calculations become critical in determining if a game is fair or not. If there was a $5 fee to play this game, then the player has the edge as they would be expected to gain $7 and have a net profit of $2. Similarly, the game would have the edge if the fee was raised to $9. A game is considered fair when the cost to play equals the expected value.
Roulette consists of a spinning wheel with the numbers 0 through 36. A ball is dropped in to the wheel, with the final position of the ball deciding the winning number. Before the spin, players can bet on the number they predict will be the winner. A bet of $10 dollars on a number would lead to winning $360 if that number is the winner. At first glance, roulette seems fair to the normal human; while the odds of winning are low, the payout is significantly larger than the original bet. But after crunching the numbers, a small edge is found. If I bet $x on a number, then I will have $36x in a winning scenario. However, due to there being 37 numbers on the wheel, I only have a 1/37 chance of winning. This means the expected value is $36x/37, just under my initial bet of $x. And slowly but surely, if you were to play roulette a thousand times, you would end up losing money over time since real averages get closer to expected value over time.
Games like roulette are favored by casinos because they seem fair to players, allowing them to conceal the edge and trick people into playing. But while expected value can help us see the biases in combinatorial games, they do not take one important fact into consideration: that the players are human. Let’s assume you had a million dollars. I now give you a choice; either keep your money or give it to me for a chance at 2.5 million or nothing based on a coin flip. A vast majority of people would take their million dollars and leave. However, the expected value tells us that playing the game is actually the better choice in terms of gaining money. What expected value does not take into account is diminishing returns. Going from broke to being a millionaire will change your life more significantly than going from 1 million to 2 million. At some point, each dollar becomes less and less meaningful because you already have so much, something that combinatorics doesn’t understand.
Expected value is a useful tool to use when analyzing the dynamics of a game. But at the end of the day, even expected value can be assisted by a little bit of common sense.