Expectations Galore: Basics

Brian Alspach

Poker Digest Vol. 2, No. 20, September 24 - October 7, 1999

Let's begin an exploration of the notion of expectation. We shall be extending the discussion to many other games in spite of the word ``poker'' in the name of my column.

Expectation has a technical definition, which we shall see soon enough, but let's first discuss it at an intuitive level. Expectation revolves around the idea of average return. Let me remind you that mathematical usage of words and colloquial usage of words frequently differ. When using the word ``return'', we allow it to be positive or negative. In colloquial English we normally call a positive return a ``win'', and a negative return a ``loss''. I am so accustomed to allowing quantities to be either positive or negative that I sometimes find myself confusing people when I talk about something being negative which most people think of only in a positive sense.

Talking about the expectation of a wager, a game or a strategy does not tell the whole story. Let's illustrate this with two examples and, as mentioned above, keep the discussion on an intuitive level. Suppose you are wagering a dollar on the roll of a single die. If it turns up 1, 2 or 3, you win a dollar, whereas, if it turns up 4, 5 or 6, you lose your bet. It is clear you have .5 chance of winning and .5 chance of losing. Thus, if you were to play this game many times in succession, you ``expect'' to win about one-half the bets and lose the remaining bets thereby making you about even. It should not surprise you that this game has expectation zero.

Now let's discuss another game. Suppose you are wagering a dollar on the roll of three dice. If the dice total three, you win $215. If the dice do not total three, you lose your bet. The probability of three dice totalling three is 1/216. Thus, out of 216 rolls of three dice, you would not be surprised to win once and lose all of the other times. If this happened, you would come out exactly even. In fact, the expectation for this game is also zero.

In spite of the fact these two games have the same expectation, it is clear your bankroll would be affected much differently in the two games. In the first game your bankroll typically is going to go up and down in a modest fashion. In the second game your bankroll is going to flucuate much more wildly. There will be long losing streaks slowly eating away your bankroll followed by a win which injects a large increase for your bankroll. A couple of wins close together would have a very positive effect on your assets.

We are not going to discuss the flucuation aspects at this time. Instead, we are going to discuss only the notion of expectation. This is not to say the flucuations are not of interest. In fact, it is a very interesting topic and should be of concern to anyone involved with games of chance. The preservation of your bankroll should be of vital concern.

The above two examples illustrate what is involved in defining expectation. There is a wager involved and a finite list of possible outcomes $O(1),
O(2),\ldots,O(m)$. Each outcome O(i) has a probability P(i) of occurring and a return of R(i) should it occur. We shall assume the amount of the wager is one unit and the return R(i) is some number of units. The expectation for this wager is then given by

\begin{displaymath}P(1)R(1)+P(2)R(2)+\cdots +P(m)R(m).\end{displaymath}

For example, in the first game mentioned above the formula produces .5(1) + .5(-1) = 0. The second game produces (215)1/216 + (-1)215/216 = 0. Thus, both games have zero expectation.

What does expectation mean in the course of putting money at risk? For an individual, expectation may be far removed from what actually happens, especially over a short time span. A lottery winner couldn't care less about the expectation of a typical lottery. Nevertheless, if an individual persists in playing some game, then in the long run his or her return will tend to conform to what is predicted by the expectation of the game.

Now look at it from a casino's viewpoint. If they have 1,000 slot machines in operation and each machine is set to return on average 5 cents for every dollar inserted, the fact one machine is in the process of giving a large jackpot to a player is somewhat irrelevant because the great majority of the machines will be in the process of extracting money from players. Thus, the player who has just hit a large jackpot will have a dramatic change in the outcome for her entire trip while the club will find their take only minutely affected. This means the notion of expectation is more meaningful when applied to many players. A casino essentially can count on producing a certain profit when the expectation is known.

In general, a player wants to exploit the rare situations in which the expectation is positive and be aware of the risk involved when playing games with negative expectation. We shall explore the expectations of a variety of games in subsequent parts of this series.

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