Let's calculate the expected value for goals that depend on more than one event. First, let's look at these two decks. Our goal is to draw a star from the white deck and a square from the black deck. What is the probability of meeting this goal? There are two stars in the white deck and one square in the black deck. The total number of possible pairs is six. So, the probability is 26ths.
Now what's the expected value? Expected value is the probability of an outcome multiplied by its payout. Here the probability is 26 and the payout is 3 points. So we multiply 2 sixth by 3.
That gives us 66 which simplifies to one point. Just like drawing a single card from a deck, the expected value of drawing a card from two decks is the probability times the payout. Let's try another goal with two independent events. Here we need to draw a triangle card and roll a number less than or equal to two. First, let's find the probability of drawing a triangle. There are three cards and only one of them is a triangle. So, the probability is 1 out of three. Next, let's find the probability of rolling a two or less on a six-sided die. Only rolling a one or two meets the goal. So, the probability is 2 out of six. To find the probability of both of these things happening, we multiply their individual probabilities together. 1/3 * 26 = 21 18. This is the probability of hitting our goal. Now let's find the expected value if the payout is 6 points. We just multiply the probability we found 21 18 by 6. 21 * 6 gives us 12 18 which is 2/3. When there are many possible outcomes, thinking about the total number of possibilities can be helpful. Let's look at a final example involving a single card draw and a coin flip. There are four possible cards and two sides of the coin. So, there are four * 2 or eight total possible outcomes. Our goal is to draw a circle and get heads on the coin flip.
Let's see how many ways we can achieve this. Two of the four cards are circles.
For each of those two circle cards, we could also get heads. So there are two ways to get our desired outcome out of eight total possibilities. This means our probability is 28. If the payout for hitting this goal is eight points, we can find the expected value by multiplying the probability 28 by 8.
This gives us an expected value of two.
When there are multiple outcomes, the best way to find expected value is multiplying the probability by the payout.
