When choosing between goals, it helps to know which one offers the best outcome.
Let's figure out which goal to choose. A star for six points or a circle for four points. There are four cards, two of which are circles and one of which is a star. To calculate expected value with a fixed reward, we multiply the probability of an outcome by the reward.
The first goal offers six points for a star. The probability of drawing a star is 1 out of four or 1/4. The expected value is 1/4 * 6, which equals 6/4s.
The second goal offers four points for a circle. Since there are four cards and two are circles, the probability of drawing a circle is 2 out of four or two/4s. The reward is four points. So the expected value of drawing a circle is 2/4 * 4 which is 8/4s. 8/4s is greater. So the circle has a better expected value.
Now the reward is the face value on the card. To calculate expected value with variable rewards, we add the possible reward amounts and divide by the total possible outcomes.
For the first goal, we add 0 + 2 + 3 + 0 and divide the sum by 4. That gives us 54s.
Next, we find the expected value for the triangle goal. That's the sum of 2 0 0 and 4 / 4. That gives 6/4s. Since 6/4s is larger than 5/4s, the triangle goal has a higher expected value. Now, let's use a six-sided die where the reward is the face value for rolling an odd number. The expected value is the sum of the odd numbers 1 3 or 5 divided by the total number of outcomes, which is six.
So, that gives us 9 sixths. For rolling a five or higher, the expected value is the sum of 5 + 6 / 6 or 116. 116ths is greater.
Even though there's a higher probability of rolling an odd number, the expected value was lower because the reward was lower.
Here we can multiply the probability of success by the reward amount. For a circle card, the probability is 2 out of 4 or 2/4s.
2/4s * 5 is 10/4s.
For an even card, the probability is 3 out of four or 3/4s. 3/4s * 3 is 9/4s.
The expected value of the circle card is higher.
Let's look at a die roll where the reward is the face value. For a roll of four or less, the outcomes are 1 2 3 and four. The expected value is the sum of those numbers divided by 6, which is 10 sixths. For a roll of five or more, the expected value is the sum of 5 and 6 / 6, which is 116ths.
Rolling a five or six has a higher expected value.
This last example shows that larger rewards can lead to a higher expected value even when they're less likely to happen. Calculating expected value can help us make the best choice.
