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Expected Value in Probability: Definition & Formula

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  • 0:03 Expected Value
  • 1:54 Probability of Each Event
  • 2:30 Formula
  • 3:18 Example
  • 6:39 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After watching this video lesson, you will understand what expected value means. You will learn how to use this value to decide whether a certain action is worth taking or not.

Expected Value

In Las Vegas, Nevada, there are a lot of casinos where you can play games for a bit of money. Each time you play these games, you can expect to win nothing, something, or a lot of money back! Of course, if you hit the jackpot and win a million dollars from just one game, you would think that these games are great! But keep playing and you will find yourself losing more and more money. These casinos are a business, and they are in it for the money just like any other business. So all the games that they offer have been mathematically and statistically scrutinized by the business team to make sure that the casino ends up making money.

How do they do this? They calculate what is called the expected value of the game. The expected value is the average value you can expect after a large number of rounds. This means that the more and more games you play in a casino, the closer you will get to the expected value for your earnings, or lack thereof. You can use this value to determine whether a game is worth playing or not. If the expected value of a game is -$5.00, for example, you might say it's not worth losing $5 to play it, or you might say that the fun of playing the game is worth losing $5.

Because this expected value is an average, you can expect to hit this number when playing the game. For example, a game of dice may have an expected value of $2, but the only choices are to win $0, $5, or $10. You will never win $2, but if you play enough rounds, you will find that your earnings per game will get closer and closer to $2.

Probability in Each Event

To calculate the expected value of a particular game, the casinos need to know the probability of each event that may happen in the game. For example, in a game of roulette, they need to know the probability or chance of the ball landing in each of the numbers on the roulette wheel. In a dice game, they need to know the probability of the dice landing on each of its numbers 1 through 6. Then, to calculate the expected value, they multiply each event with its probability and sum it all up. This gives them the expected value of that particular game.

Formula

We can write this calculation in formula form by using the summation symbol like this:


expected value


In this formula, x represents our event. So this formula is telling us to multiply our event with its probability, and then sum all these multiplications together. For example, if the probability of rolling a 5 in a dice game and winning $10 is 1/6, we would multiply the $10 (the event) with 1/6 (the probability of it happening), and then add it to all our other choices.

In a game of dice, since there are only six possible events, we only have to add up 6 events. For the roulette wheel, since there are 38 choices, we will need to add up 38 events.

Example

Let's look at an example:

It costs $3 to play a game of dice. You can expect to win $5 if you roll a 5 and $10 if you roll a 6. What is the expected value of this game? This is in addition to winning your $3 back, so a net win of $5 or $10.

We first need to make a table of our events and the probability of it happening. We know that the probability of rolling each number on the dice is 1/6 for each number. So we can write a table listing our dice numbers, the money we earn when we roll that number, and the probability of it happening.

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