Expectation, or Expected Value, is simply a weighted average, a way to calculate risk versus reward. We tend to think this way naturally, without numbers, but simply assessing whether taking a risk is worth it. For example, if you speed a little on the freeway, your chances are low of being ticketed, and the fine may not be much, so you feel safe speeding a little (note: we here at www.surgent.net do not condone speeding, especially really fast speeding). However, if the probability of being caught was still the same, but the sanction was six months in jail, then the risk isn't worth it, and you probably would not speed.

The general structure of finding an Expectation "E" is:

E = (probability of winning)(amount you win) + (probability of losing)(amount you lose)

The "amount you lose" is stated as a negative number, and "winning" and "losing" can be translated to fit the scenario.

Easy example: a single die has six faces. The probability of any one face coming up on a given roll is 1/6, so the probability of a particular face not coming up is 5/6. Suppose you pay $1 to roll the die once. If it comes up a 6, you win $4, minus the dollar you bet, so you actually win a net $3. Otherwise, you just lose your dollar. The expectation of this game is

E = (1/6)($3) + (5/6)(-$1) = 3/6 - 5/6 = -2/6, or -$0.33.

The expectation is negative. In this game, you lose on average 33 cents per dollar played. In the long term, you will lose. If you played a hundred games, you would have bet $100, but would only have about $67 now. You are slowly losing your money. In other words, the occasional wins do not cover your losses.

Suppose this same game is played but you win $10 if 6 comes up (thus, you actually net $9 in winnings). The expectation is now

E = (1/6)($9) + (5/6)(-$1) = 9/6 - 5/6 = 4/6, or +$0.67.

Now, your expectation is positive! You win, on average, 67 cents per dollar bet. If you played 100 games, betting $100, you should be up about $67.

Let's say you win $6 if 6 comes up (so you net $5 per win), your expectation is

E = (1/6)($5) + (5/6)(-$1) = 5/6 - 5/6 = 0.

The expectation is 0. This means, in the long term, your wins will cover your losses exactly. If you played a hundred games, betting $100, then you would be have about $100. You wouldn't be winning or losing in the long term. Expectations that are 0 are called "fair games".

Math odds vs House odds

The winnings per game are simply what the house decides to pay. These are different from the odds calculated using mathematics. The math odds are always what would be paid out to keep a game fair. The house odds are always less than the math odds. If a game has 10:1 math odds, then you should net $10 per dollar bet every time you win, and in the long term, would not win or lose. You would be at a steady state. However, the house (the casino) knows this, so they simply ratchet down the house odds to something like 7:1, meaning you net $7 per dollar, while the house keeps that extra $3, which helps pay for all the glitzy lights and lounge acts.

Odds is just a different way to state a probability. In the above example, the probability of a 6 coming up is 1/6, so the probability against a 6 coming up is 5/6. The math odds are then 5:1, meaning 5 losses per (:) 1 win. Gambling odds are always stated "loss:win". Thus, the math odds of this game is always 5:1. The house then decides its own odds (its payout odds). In the first case, the house odds were 3:1, so the house always wins in the long term. In the second case, the house odds were 9:1, so the player always wins in the long term. In the third case, the house odds were 5:1, matching the math odds, so no one wins in the long term. Everything stays balanced.

It should come as no surprise that house odds are always less than the math odds. All games of chance at any casino worldwide are designed so that the player's expectation is negative.

People may grumble about some elaborate scheme by which the casino governs its payouts, even assuming there's some funny stuff with the actual machines or the card dealers. The truth is, it's shockingly simple and mundane. The casino doesn't need to do much. They can hire the smartest mathematicians who can calculate the math odds for any game of chance, then the casino just needs to pay less than the math odds. It's that simple. Occasionally, someone wins big, maybe a million dollars. But the house has accrued so much in its coffers by the losses of the collective mass of other players that paying out the million does not affect their overall strategy.

One must be very careful to leave the emotional aspect out of gambling. The house understands the psychological pull that gambling has on people. Games are designed to cause players to think "they almost won". There's no such thing as "almost winning". You either win or lose, and anything suggesting "almost" is an illusion meant to keep your spirits up and to continue to feed the machine the coins.

How fast do I lose?

Expectation is a good way to gauge how "fast" one loses by playing the game repeatedly. The closer to 0 the expectation, the slower one loses. Similarly, the farther away from 0, the "faster" one loses. For the sake of ease, this can always be stated per dollar played. If you bet $10 or $100 at once, your expectation value won't change in relative terms. It'll just be proportional, that's all.

Video poker has theoretically 0 expectation, but this assumes one plays it exactly right all the time. Otherwise, the expectation will be slightly below 0. People skilled at video poker such as the wife of the writer of this page can "stay close", and the occasional big win can cover the losses if the player is smart enough to walk away. The wife of the author of this page has the smarts to do so, and we have enjoyed many a free meal as a result. But the psychology of winning can overwhelm someone, where they'll be fooled into thinking that they did something special to win, that they're on a good-luck roll. So what do these chumps do? They play more and more... and simply lose that money they have won. Allow me to repeat myself: you will never ever beat the house at the house odds if you play long enough. If you happen to win big, stop and step away.

Craps has an expectation of about -$0.015, or about a loss of one and a half cents per dollar played. Roulette has an expectation of about -$0.05, or about the loss of a nickel per dollar played. These games allow one to "stay close" to breakeven over a longer period of time. When the occasional big win happens, it may actually cover one's losses and then some, and that's when you step away. Expectation assumes you play "forever". In real life, this is limited to how much money you have available to play.

Slots have awful expectations, about -$0.40 to -$0.50 per dollar. You lose very fast. Keno tends to be about -$0.30 per dollar played. You lose fast at Keno too, but what makes Keno so devious is the way the board is displayed, in an 8x10 grid of numbers. So you select a number as part of your bet, then see the board light up with numbers "surrounding" your number, and you think "man, if I'd bet just one number lower, I'd've won. I must play some more because I'm so close...". No such thing. You did not almost win. No one was deliberately selecting numbers "close" to yours. You simply obeyed the laws of probability without trying.

The author of this page understands that gambling is an addiction to some people and that they probably understand that it's a fool's errand, but the emotional hit of winning is so strong they keep doing it. It is imperative to remove any emotional aspect to gambling. You don't win because you're special or lucky, or on a hot streak, or anything like that. The casinos know how to tweak that feeling to their advantage. For example, let's say you win and then stop playing. Now what. There's not much else to do in a casino except gamble, eat, drink or buy stuff at the souvenir store. They're going to get back your money in some manner. The worst customer to a casino is the person who wins then walks out the door. The author of this page has bet $10 and by blindly obeying the laws of probability, has occasionally seen his money grow to $12, a 20% increase in interest. He then walks away. 20% is an excellent rate of return, whether on $10 or $1000. The author of this page also doesn't really like gambling. He likes the atmosphere, the buffet meals, and looking at all the people of various levels of tackiness. He enjoys playing Keno, knowing his odds are lousy, but since Keno games ("races") happen every five minutes or so, his rate of loss per unit of time is slowed, allowing him a chance to wander to the sportsbook and watch eighteen games and horse races at once, and time to see how his lovely wife is winning at video poker so we can go and eat for free.




By Scott Surgent. Please send feedback or error notification to me at scott dot surgent at gmail. Updated 10/27/22.