At a carnival, you spin a spinner that has four equal-sized sections, each a different color: green, yellow, red, and blue. If you land on green, you win 2 points. If you land on yellow, red, or blue, you lose 1 point. Write the expected value equation.
E = 2(1/4) - 3(1/4) = -1/4
gjg
The expected value equation is calculated by multiplying the probability of each outcome by its associated value, and then summing up all the products.
Let's denote the probabilities of landing on each color as follows:
P(green) = p(G)
P(yellow) = p(Y)
P(red) = p(R)
P(blue) = p(B)
The corresponding values for each outcome are:
Green = 2
Yellow = -1
Red = -1
Blue = -1
The expected value equation can be written as:
Expected value = (P(G) * 2) + (P(Y) * -1) + (P(R) * -1) + (P(B) * -1)
So, the expected value equation for this scenario is:
Expected value = (p(G) * 2) + (p(Y) * -1) + (p(R) * -1) + (p(B) * -1)
To write the expected value equation, we need to calculate the expected value of each outcome and multiply it by the probability of that outcome occurring.
Let's calculate the expected value for each possible outcome.
1. Landing on green: You win 2 points. The probability of landing on green is 1/4 (since there are four equal-sized sections). So, the expected value for landing on green is (2 points) * (1/4 probability) = 1/2.
2. Landing on yellow, red, or blue: You lose 1 point. Since each color has an equal-sized section on the spinner, the probability of landing on yellow, red, or blue is (3/4) * (1/3) = 1/4. Therefore, the expected value for landing on yellow, red, or blue is (-1 point) * (1/4 probability) = -1/4.
Now, let's sum up the expected values for each outcome to find the expected value of the carnival game:
Expected Value = (Expected value for landing on green) + (Expected value for landing on yellow, red, or blue)
Expected Value = (1/2) + (-1/4)
Expected Value = 1/2 - 1/4
Expected Value = 2/4 - 1/4
Expected Value = 1/4
Therefore, the expected value of the carnival game is 1/4, or 0.25 points.