# Expected Value

## Expected Value

Hello and welcome to this video about expected value! In this video, we will explore what expected value is and how it can be used.

Expected value is an application of **probability** that is typically used to help people make decisions about the **viability of risks,** most often financial risks. The big idea is to multiply the probability of each **potential outcome** of an event by the value of that outcome and add all the results together in order to make a more informed decision.

Let’s begin by talking about coins. The **probability** of flipping a coin and it landing on heads is 1 out of 2, as is the probability of landing on tails. Let’s say that you are thinking about playing a game that costs $1.00 to enter. The game will pay you $2.00 if the coin lands on heads and $0.00 if the coin lands on tails. Let’s make a table. Here we have heads and tails and on the left we have the value of the outcome and the probability of the outcome. If you win the game, you get $2.00, but since you pay $1.00 to play in the first place, we subtract that from the $2.00, which gives us $1.00. This means that if the coin lands on heads, you are one dollar richer than before you played. Now, if the coin lands on tails, you win no money and we still have to subtract that one dollar we spent to play the game, so we come away with one less dollar than we started out with. And like we mentioned before, a coin has two sides, so the probability of the outcome in both cases is 1 out of 2.

**To calculate the expected value,** we’ll use this formula: the expected value of x equals the probability of outcome 1 times the value of outcome 1 plus the probability of outcome 2 times the value of outcome 2.

If we plug in our numbers, our calculation looks like this:

An expected value of $0.00 illustrates that the probabilities of the event outcomes and the values of those outcomes are equal, as is shown in the table. The game is balanced.

You decide not to try the coin game another time and instead head over to the movie theater. On the concession counter is a drawing that costs $1.00 to enter. The first entry drawn wins 2 movie tickets (a $30.00 value), the second entry drawn wins a large popcorn and drink (a $15.00 value) and the third ticket drawn wins a large popcorn (an $8.00 value). You ask the manager how many entries they tend to get for drawings and he says about 100. Do you take the chance? Let’s fill in our table, looking at each prize individually:

You know that drawings are stacked against you in the first place, but since the expected value shows a loss that is less than half of your cost to enter, the risk is acceptable. You decide to take the plunge.

After the movie you head home. You check your mail and find an ad for bundling TV, phone, and internet in one package using a service that isn’t even available in your area. The ad is elaborate, high-quality, and beautifully printed on glossy cardstock. You wonder how the company makes any money sending these costly ads to places that can’t even take advantage of them.

The company knows that they’ll get about a 1% return on their mail investment. In other words, they’ll get one long-term customer for every one hundred ads that are sent. If a long-term customer is worth at least $3600 and each ad costs $5.00 to print and mail, is sending direct mail a profitable idea for the company if they send 10,000 ads?

On average, the company can expect to gain $31.00 for each ad that is sent, which is a very lucrative decision!

You throw the ad in the trash and begin looking forward to the outdoor festival you are planning to attend the next day. You check to see what the weather will be like and discover a 75% chance of rain. You remember an article you read describing how $25,000 was spent by the city to put the festival together. Tickets cost $10.00 apiece and the expected turnout is about 8500 people. Is it still a good idea to put on the festival, given the rain chances? Let’s map it out and see:

With these rain chances, the expected value of the city’s investment is -$3750. They may want to postpone the festival for a sunnier day.

Thanks for watching this overview of expected value! I’ll see you next time.