Let's compare different rewards using expected value. In this first scenario, we're drawing one card each from two decks. The goal is to draw a circle card from the white deck and a card with a value of two or more from the black deck. We can calculate expected value by multiplying the probability of the event by its reward value. There are two circle cards in the white deck and two cards two or more in the black deck. So that gives us four successful pairs that meet the goal. In total, there are 3 * 2 or six possible pairs. So the probability is 4 sixths. Then we multiply by 1. So the expected value for this goal is 46th. Let's find the expected value for a different goal using the same two decks of cards. This goal gives five points as a reward.
Again, there are six possible pairs.
Only one pair, a white square and a card equal to three meets the goal. The expected value is 16 * 5 points, which equals 56.
Now let's compare the goals we just calculated expected value for. For the first goal, we found the expected value to be 46ths. For the second, we found it to be 56ths. So even though the probability of the second goal is smaller, it has a higher expected value and is the more valuable goal. Let's try another comparison. We draw one card from each of these two new decks. The goal gives us six points if the sum of the drawn cards equals six. Let's find the probability first. There are 3 * 3 or nine possible outcomes in total. Then there is only one pair that sums to six, a white three and a black three. The reward is six points. So the expected value is 1 9th * 6 or 6 9th.
Using that same deck, let's analyze a different goal. This time, we get four points for drawing two cards that sum to four. The probability is again out of nine. To meet the goal, we could draw a white one and a black three, a white three and a black one, or a white two and a black two. So, the probability is 3 out of nine. Then the reward is four points. So, 3 9th * 4 is 12 9ths. Now we can compare the two goals we just calculated. The first had an expected value of 6 9ths while the second had an expected value of 12 9ths. The second goal has a higher expected value so it is more valuable on average.
This is how calculating expected value helps us make better decisions. We can quantify the potential of different choices and pick the one that offers the highest expected value.
