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Expected Value

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In this tutorial, you're going to learn about the expected value of a probability distribution. Expected value can sometimes be a confusing term. So I like to also use the term mean or average of a probability distribution.

So let's start with an example. Let's take a look. Consider this spinner has three 1's, a 2, two 3's, and two 4's. The probability distribution is shown here. The outcomes are 1, 2, 3, and 4. And their probability are 3/8, 1/8, 2/8, and 2/8, respectively.

Now, suppose we were trying to determine an expected value, or what we would expect to happen. Well, we're going to talk about expected value as a long-term average value that this spinner might produce. So let's think about spinning it a bunch of times. What do you think would happen?

Well, suppose this spinner was spun eight times. Suppose it went according to our expectations. Well, if we spun it eight times, I guess our expectation would be for it to fall on each sector once. It doesn't have to happen that way. But it seems like that would be what we expect to happen.

So we would obtain three 1's, one 2, two 3's, and two 4's. And the law of large numbers says that the longer we do this, the long term, this is exactly what will happen. About three out of every eight times, you'll get a 1. About one out of every eight times, you'll get a 2, et cetera.

So we're going for long-term average. And the law of large numbers says that this will happen in the long-term. Now, if that's the case, the mean is just going to be the average value from those eight spins, which is 2.375.

Now, let's look back to how that was calculated. So we did this fraction. But you can also rewrite that fraction by grouping the 1's, grouping the 2's, grouping the 3's, and grouping the 4's. So the 1 appeared three times. The 2 appeared one time. The 3 appeared twice. And the 4 appeared twice.

We can continue breaking this down by saying, really, all of these addends, all of these terms here, are in fact individually divided by 8 and then added. So now it might look like that. Now we start to look at 1, 3, and 8, 1, 2, and 8, et cetera.

Some of these look familiar. In fact, this fraction can be rewritten a different way. All of these fractions can be rewritten with this item in the parentheses sort of out in front or out behind, like this. 1-- pull it out front. And 3/8 is left. 2-- pull it out front. 1/8 is left there.

Now look at these. These numbers are looking awfully familiar. It seems like these were the numbers from the probability distribution, where these numbers-- 1, 2, 3, and 4-- were the potential outcomes, the values we're going to call x. And these fractions-- the 3/8, 1/8, 2/8, and 2/8-- were their probabilities.

So it looks like what we're doing is multiplying each outcome times its probability and then doing that again-- outcome times probability and then outcome times probability-- and adding them all together. And so this is the way we're going to calculate expected value, the sum of each outcome times its individual probability.

So the expected value, also called the mean of a probability distribution, is found by adding the products of each individual value, each outcome, and its probability. Now, in our case, we ended up with 2.375. Now, you can't get 2.375 on the spinner. And it's not even an integer.

The thing is because we're talking about long-term average, it doesn't actually need to be an integer. And it doesn't need to be possible, either. And then one additional note-- notationally, we can write expected value of a probability distribution for x as mu of x-- mu meaning mean-- or E of X-- E meaning expected value. Both of these are accepted notations. Typically, I'm going to be using the mu.

So try this one on your own-- the mean of a distribution showing the payouts and probabilities from betting red on a roulette wheel. Now there are 18 red sectors on a roulette wheel. And when you win, you win $1 if you had bet $1.

There are 20, therefore, that aren't red out of the 38. All the sectors are equally likely. And if you don't win, you lose the dollar that you had put in. So find the expected value for a play on the roulette wheel. Go ahead and pause the video to figure it out.

What you should have come up with was this. The mean of the wheel is equal to negative 1 times its probability. And the other outcome is you gain a dollar-- positive 1 times its probability. When you solve it, you end up with negative 2/38, which is about negative 0.05.

What that means is over the long-term, you lose about a nickel every time you play $1 bet on a roulette wheel. Now, that's not much. You can sit at the roulette wheel, therefore, for a while and continue to play. But because it's negative, that means you will be losing money over the long-term.

And so to recap, the expected value, also called the mean, of a probability distribution is taken by multiplying the probability times its outcome. And then you add each of those products. It doesn't have to be an integer. It doesn't have to be a possible value. So we talked about expected value, also called the mean. Good luck. We'll see you next time.