Expectation

One of the most important concepts in sports as well as probability deals with the concept of expectation. Statistics such as ERA or points per game are attempts to use expectation to quantify how good a player or team is. Although these stats aren't perfect, they are based on the concept of expectation.

In a nutshell, the expectation of a random variable is the average value. If there are only a finite number of possible outcomes, the technique for finding the expectation is simple. We take the sums of all possible outcomes times the probability of that outcome. For instance, if we're rolling a balanced 6-sided die the expectation of the number that we roll is

1/6*1 + 1/6*2 + 1/6*3 + 1/6*4 + 1/6*5 + 1/6*6 = 3.5

This technique works even if not all the probabilities are equal. Suppose we have a coin that falls on head 75% of the time. We want to know the expected number of heads if we flip the coin once. This gives E(Heads) = 0*.25 + 1*.75 = 75%.

If we are dealing with a situation in which there are an infinite number of possible outcomes, it may still be possible to calculate expectations in this way. This would require doing an infinite summation, which unfortunately means that our expectation may also be infinite.

Linearity of Expectation

One of the important properties of expectation is that it is linear. This means that if X and Y are random variables with finite expectations and a and b are constants, then E(aX+bY)=aE(X) + bE(Y).

Suppose we have a red die which we'll call R and a green die which we'll call G. We want to know the expectation of 3R+5G. Computing this expectation manually would be tough since we'd have to consider all possible outcomes. However, using the linearity of expectation it becomes easy. We know that E(G)=E(R)=3.5 and so E(3R+5G) = 3E(R)+5E(G)=8*3.5=28.

Another example is if we flip a balanced coin 100 times and want to know the expected number of heads. Calculating this directly would be extremely difficult, but using the fact that we know this expectation is 1/2 for a single coin, the expectation is 100*1/2=50.

Expectation and Gambling

If you want to become a successful gambler, the most important question you will need to ask yourself before making any bet is "What is the expectation of this bet?" (Ok, that last statement isn't entirely true based on practical conditions, but assuming you have a large enough bankroll, it's basically the only question that matters). If the expectation of a bet is greater than zero, it is said to have positive expectation (also denoted +EV). For instance, suppose I make a bet with SBG saying that we will flip a coin and he will pay me $100 if the coin is heads and I will pay him $105 if it is tails. Half of the time SBG will lose $100 and half the time he will win $105. So the expectation for SBG in this bet is .5*$105 + .5*(-$100) = $2.50. This bet is +EV for SBG and so he should take it (assuming he can afford the $100 if he loses). From my perspective, the bet is -EV and so I should avoid it unless I figure I will gain at least $2.50 in entertainment from the wager.

One common misconception about sports bettors is that they need to be able to predict winners with incredibly accuracy. This is not necessarily true. In fact, a sports bettor can lose more than half of his bets and still make money. Consider a baseball game between the Red Sox and the Royals. The Red Sox would likely be heavily favored to win this game and so a sports book might offer you 2:1 odds if you're willing to bet on the Royals. You have done extensive mathematical calculations and concluded that the Royals will win 40% of the time. If you wager $1, you will lose $1 60% of the time and win $2 40% of the time. This gives an expectation of .6*(-1)+.4*2=20 cents. Even though you will be losing money 60% of the time, you will be making 20 cents on the dollar for every. Note also that due to the linearity of expectation, you will win 40 cents on a $2 bet, $2 on a $10 bet, $200 on a $1000 bet, etc.

Sports and Expectation

This topic deserves its own post, so I will save it for later.

Problems

1. A typical sports book will offer -110 lines for things such as spread bets and totals. This means that in order for you to bet on these lines, you will have to wager $110 to win $1000 (these need not be actual amounts, you could wager $55 to win $50 or $1100 to win $100 for instance). Assuming there is a 50% chance of either side winning, how much will the sports book will the sports book make on average for a $100 bet (This quantity is called the vig)? How frequently does your side need to win in order to make this a profitable bet for you?

2. In role-playing games, you will frequently roll 4 balance 6-sided dice and compute the sum of the largest three (for instance, if you roll 2,3,5,6 your total will be 14. If you roll 4,4,4,4 your total will be 12). What is the expected value of this sum?

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