What the heck is Expected Value?

It is exactly what you would think it is. If you conduct an experiment, the expected value is how many times a certain event is expected to happen. For example, if you roll a die 6 times, how many times do you expect to roll a 2?

 

How to Calculate Expected Value

The probability of each event is the same:

Suppose you conduct an experiment with n trials, where each event has probability p of occurring. Then the number of times we expect the event to occur is np.

Take our example above: if you roll a die 6 times, how many times do you expect to roll a 2?  Since each event has a probability of p = 1/6 of occurring and there are n = 6 trials, you would expect to roll a 2: np = 6(1/6) = 1 time.

Notes:

  • The expected value for the random variable X is usually denoted as E(X).
  • The expected value is also called the mean, which is denoted by the Greek letter \mu.

The probability of each event is different

Suppose you conduct an experiment with the random variable X,with sample space { x_{1}, x_{2}, x_{3}, \ \text{...} \ , \ x_{n}}} }, with corresponding probabilities { p_{1}, p_{2}, p_{3}, \ \text{...} \ , \ p_{n}}} }, where P(X = x_{i}) = p_{i}  for i = 1, 2, 3, …, n. Then the expected value of the random variable X is:$$E(X)=\mu = \sum^n_{i=1}x_ip_i \ \ \ \ \ \ \ \ \text{or} \ \ \sum^n_{i=1}x_iP(X=x_i)$$

 

Fair Games

This is for all of the gamblers and game players out there. If you play a game where you can win and lose money, typically you will want to know if a game is “fair”.

A fair game is a game in which the expected value is zero. ie: E(X) = 0.

In casinos, NO GAMES ARE FAIR. The expected value of casino games are always negative!! This is how casinos make money. Never go to casinos to make money. Know your limit, play within it 😉.

If you lose money, the value of x_i is negative. If you gain money, the value is positive. This will make more sense with an example.

Example:

You pay $1 to play a game where you flip a coin. If the coin lands on a heads, you lose $0.50. If the coin lands on a tails, you win $2. Is this a fair game?

Since you need to pay to play the game, you lose $1 regardless of if you win or lose. The probability of rolling a heads or tails is 1/2 each. So the expected value of this game is:

E(X) = -1 + (-0.5)(1/2) + 2(1/2) = -1 – 1/4 + 1 = -1/4 = -0.25

Since the expected value is not equal to 0, this game is not fair. In fact, if you were to play this game, you are expected to lose money.