Expected Value
Each session of play at a casino entails a certain amount of risk. In general, casino games are structured so that a low degree of risk usually results in small returns on investment or relatively harmless losses, while a greater level of risk may bring about some spectacular successes and failures. Players will typically wager only on games in which the potential rewards seem to warrant the risk required. They must therefore have some idea of how much can be won or lost relative to the cost of play—what gamblers call an “Expected Value” (EV).
Calculating EV
One way to think of EV is as a player’s predicted future gain or loss, or a representation of how much one expects to win or lose, if bets with identical odds are repeated many times. Mathematically speaking, EV is “the sum of the probability of each possible outcome of a wager multiplied by its payoff (value).”
A simple example of this is a coin toss in which calling the outcome correctly, heads or tails, results in a payout of 1-to-1 or “even money.” No matter whether the player calls heads or tails, the EV will always be the same. There are only two possible outcomes, each with the same probability of occurrence (1 in 2). The player can thus expect to be correct half of the time and wrong half of the time, so where “B” is the amount bet
EV = ½ (B) + ½ (-B) = 0.
In other words, the player’s forecasts result in an Expected Value of zero. Over the course of many, many flips, the number of heads occurring should be roughly equal to the number of tails, and the player can expect to break even, neither winning nor losing.
Any game of this sort, in which the player’s EV is zero (no net gain or loss), is called a “fair game.” Note that “fair” in this case does not refer to the game’s technical process but to the House and the player having exactly the same likelihood of winning/losing.
Applying EV to Casino Games
According to this definition of “fair game,” the vast majority of gambling opportunities offered by casinos are not truly “fair.” They do not give the player an equal chance of winning. Instead, they favor the House and result in a negative EV for the player.
More specifically, over the long term, the player’s EV for any game or given wager will tend to approximate the House Edge. In a game like American Roulette, where the House has a built-in advantage of 2-in-38 on every spin, the EV for the player is -0.0526. That means the player can expect to lose roughly a nickel on every dollar wagered. For Baccarat, the House Edge is 1.06%~1.24%, so the player can plan on losing about penny out of every dollar.
However, casino games are not simply pure applications of probability calculus. Calculations of long-term results are not always reflected in short-term outcomes. Plus there is a human factor involved, and the progress of many games can be influenced by the action of the players.
Blackjack is a case in point. Each choice to hit, stand, double down, split, surrender or take insurance comes with its own Expected Value. Players who master basic strategy greatly improve their EV in general, and a card counter at the Blackjack table can often identify situations where the House Edge is negative and the EV is positive.
In fact, the House has an advantage of 4% or more for about one third of all Blackjack deals, while the player has a positive EV of 0.04 or higher for another third of the hands. By skillfully increasing or decreasing one’s bets at the appropriate times, it is possible to win with mathematical certitude in the short term.
Likewise, Video Poker can offer a positive EV under certain circumstances. It has been shown that the game called “Full Pay Deuces Wild” gives the player a positive EV of 0.0076 when the optimum card selection strategy is applied. The same can be said of playing progressive slot games only when the jackpot is big enough or knowing when to raise in a game of Caribbean Stud. This is where human skill overcomes the randomness inherent in games of chance, improving the player’s Expected Value.
