This tutorial will discuss determining expected value by focusing on:
Expected value can sometimes be a confusing term. So I like to also use the term mean or average of a probability distribution.
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.
When trying to determine an expected value (what be expected to happen), expected expected value will be discussed as a long-term average value that this spinner might produce.
What do you think would happen if someone spun it a bunch of times?
Suppose this spinner was spun eight times and it went according to the expectations. If spun eight times, he 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 the expectation.
So three 1's, one 2, two 3's, and two 4's are obtained. The law of large numbers says that the longer this is done, 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, etc.
So you'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.
How was that calculated? By solving this fraction:
However, 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.
You can continue breaking this down by having each addend individually divided by 8 and then added. So now it might look like this:
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 1 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 that will be referred to as "x". And the fractions-- the 3/8, 1/8, 2/8, and 2/8-- were their probabilities.
So this is what's happening: multiplying each outcome times its probability and then doing that again-- outcome times probability and then outcome times probability-- and adding them all together.
This is how expected value will be calculated: the sum of each outcome times its individual probability.
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. In this case, 2.375 was the result. It is not a possibility on the spinner, nor is it an integer.
Since the focus is about long-term average, it doesn't actually need to be an integer. And it doesn't need to be possible, either.
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, mu will be used:
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. Find the expected value for a play on the roulette wheel.
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 that over the long-term, you lose about a nickel every time you play $1 bet on a roulette wheel. You can sit at the roulette wheel, but because it's negative, that means you will be losing money over the long-term.
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.
Source: This work is adapted from Sophia author jonathan osters.
The long-term average value taken by the outcomes from a chance experiment. It does not need to be one of the possible outcomes.