Let's calculate some expected values.
Think of expected value as the average result you'd get if you played a game over and over again. In this first scenario, we have four cards. Our goal is to draw a square, which we get four points for. To find the expected value, we find the average of all the outcomes.
Our possible outcomes are four points for the first card, zero points for the second, four points for the third, and zero points for the fourth. So the average is the sum of our outcomes 4 + 4 divided by the number of outcomes. That gives us 8 / 4. The expected value of drawing a square is 2.
Let's look at a different one. We have three cards. The goal is to draw a triangle. And if we do, we get three points. If we draw either of the other two cards, we get zero points. To find the expected value, we average all possible outcomes. That gives us three points plus 0 plus 0. Then we divide by three possible outcomes. 3 / 3 equals 1.
So the expected value of this goal is 1.
For fixed reward goals, we can calculate the expected value as the probability of meeting the goal multiplied by the reward. Let's apply this idea. Here we have three cards. Our goal is to draw a square for which we get three points.
The probability of drawing a square is two out of three or 2/3.
The reward for drawing a square is three points. So we can multiply 2/3 by three which gives us two. So the expected value is two.
Here's a different scenario. The goal is to draw a card with a value of two or more. We have three cards numbered 1, two, and four. To find the expected value, we can find the probability of meeting the goal and multiply it by the reward. 2 and 4 are greater than or equal to 2. So, the probability is 2 out of 3 or 2/3. We then multiply 2/3 by the value of the reward four.
We learned a new way to calculate expected value when the reward amount is fixed.
