# How to Calculate Expected Value Probability

Whenever I think about expected value in probability, I always remember my professor in college talking about a fair coin. A fair coin is a coin that gives equal probability for either a head or tail to occur when you flip it.

For this hub, I will be explaining what expected value is, how to calculate it, and show an example by calculating the expected value of a betting game involving a fair coin.

## About the Author

Dez was a mathlete in high school and has a master's degree in Applied Mathematics from one of the best schools in the Philippines.

## What is Expected Value Probability?

Based on the literal meaning of the words, it is basically the value you expect to get should you do an experiment whose outcome is represented by the random variable. It uses probability to find out what the expected payoff will be.

In a more formal terms, it is the weighted average of all possible values that a random variable can take on based on the terms given for the experiment. It is the average of all possible outcomes, adjusted for the likelihood that each outcome will occur.

## History of Expected Value Probability

I find that learning the history of a mathematical process helps me remember the uses of it better than just blindly memorizing it. According to Wikipedia, expected value probability originated because of this problem: "How do you divide the stakes in a fair way between two people who have to end their game before it is properly finished?"

Blaise Pascal and Pierre de Fermat have exchanged letters trying to solve this problem. They came up with this basic principle: "The value of a future gain should be directly proportional to the chance of getting it." They weren't able to publish this, though. It was only until Christian Huygens published the principle in his book that people learned of this expected value probability. He also added more complicated situations and showed how to calculate their expectations (e.g. having more than two players, more than two possible outcomes, etc.)

## Formulas for Expected Value Probability

First, let's look at a discrete random variable. What is a discrete random variable? Basically, it means that the random variable is countable. Examples include the number of students in a classroom, the number of eggs in a basket, etc.

So, suppose your random variable is X, which can take a value of x_{1} with probability p_{1}, x_{2} with probability p_{2}, and so on. The formula for the expected value probability of X is:

This formula works for both the finite case and the countable case. A countable set can be counted, but may never actually finish (infinite case). However, in order to find the expected value for an infinite countable set, the series should converge absolutely. This means that if you add the series of infinite numbers, it should have a finite solution.

In the case of a continuous random variable, where the random variable can assume any numerical value in a given interval, the formula uses the integral function:

## Finding the Expected Value Probability for a Discrete Random Variable

In order to make this hub more user-friendly, I will only be showing the steps to solve for the discrete (finite) case. This is a very applicable concept we can use in simple things we encounter in our normal lives.

**Steps:**

- Figure out the probabilities and their corresponding outcomes for the problem at hand.
- Multiply them together to find the outcome for each probability.
- Add all the products and you will get the expected value.

## Example (Fair Coin)

Suppose you are betting on a fair coin. A fair coin has equal probability that a head or tail will be on the face of the coin when flipped. You will win $10 if it lands on a head, and get nothing if it lands on tails.

Following the steps, let X be the payout and P represent the probability. For heads, X_{H} = $10 with P_{H} = 0.50. For tails, X_{T} = $0 with P_{T} = 0.50.

Then, we multiply the X's with their corresponding P's:

X_{H} * P_{H} = $10 * 0.50 = $5.

X_{T} * P_{T} = $0 * 0.50 = $0.

Then, adding the products, we get E(X) = $5 + $0 = $5.

Thus, our expected value (of payout) is $5.

If we are asked to pay a fee before being allowed to bet, we would only be willing to pay a fee that is less than the expected payout of $5. If the fee is indeed $5, this is called a *fair game*. This just means that neither side has the advantage over the other, and the expected value will then be 0.

How did it become 0? You follow the same steps, but when adding the products together, you need to take into account that you paid $5 at the start of the game, so

E(X) = $5 + $0 - $5 (for the betting fee) = $0.

## Practical Uses of Expected Value Probability

Based on the example, expected value probability actually gives you the amount you should be willing to pay to join a game involving probabilities. This may include joining the lottery, betting on card games, or even as simple as a game involving flipping coins.

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