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.
[tex]
\begin{array}{l}
E(V)=\frac{1}{a}(2)+\frac{1}{b}(c)+\frac{1}{d}(e)+\frac{1}{f}(g) \\
a=\square b=\square c=\square b=\square b=\square \\
d=\square b=\square b=\square \\
g=\square b=\square
\end{array}
[/tex]
Answer (1)
The probability of landing on each color (green, yellow, red, blue) is 4 1 .
The expected value equation is formed by multiplying each outcome by its probability and summing the results: E ( V ) = 4 1 ( 2 ) + 4 1 ( − 1 ) + 4 1 ( − 1 ) + 4 1 ( − 1 ) .
The values for the variables in the equation are: a = 4 , b = 4 , c = − 1 , d = 4 , e = − 1 , f = 4 , g = − 1 .
The final expected value equation with identified variables is: E ( V ) = 4 1 ( 2 ) + 4 1 ( − 1 ) + 4 1 ( − 1 ) + 4 1 ( − 1 )
Explanation
Analyze the problem We are given a spinner with four equal-sized sections: green, yellow, red, and blue. Landing on green yields a win of 2 points, while landing on yellow, red, or blue results in a loss of 1 point. Our goal is to write the expected value equation for this scenario.
Determine probabilities Since the spinner has four equal-sized sections, the probability of landing on each color is 4 1 .
Write the expected value equation The expected value, denoted as E ( V ) , is calculated as the sum of each outcome multiplied by its probability. In this case: E ( V ) = P ( g ree n ) × ( 2 ) + P ( ye ll o w ) × ( − 1 ) + P ( re d ) × ( − 1 ) + P ( b l u e ) × ( − 1 ) where:
P ( g ree n ) = 4 1
P ( ye ll o w ) = 4 1
P ( re d ) = 4 1
P ( b l u e ) = 4 1
Substitute the probabilities and identify variables Substituting the probabilities into the equation, we get: E ( V ) = 4 1 ( 2 ) + 4 1 ( − 1 ) + 4 1 ( − 1 ) + 4 1 ( − 1 ) This matches the form: E ( V ) = a 1 ( 2 ) + b 1 ( c ) + d 1 ( e ) + f 1 ( g ) where a = 4 , b = 4 , c = − 1 , d = 4 , e = − 1 , f = 4 , g = − 1 .
State the final equation and variable values Therefore, the expected value equation is: E ( V ) = 4 1 ( 2 ) + 4 1 ( − 1 ) + 4 1 ( − 1 ) + 4 1 ( − 1 ) and the values for the variables are: a = 4 b = 4 c = − 1 d = 4 e = − 1 f = 4 g = − 1
Examples
Expected value calculations are used extensively in the insurance industry to determine premiums. Insurance companies assess the probability of different events (like accidents or illnesses) occurring and the associated costs. By calculating the expected payout, they can set premiums that allow them to cover potential claims and remain profitable. This ensures that the insurance company can meet its financial obligations while providing coverage to its customers.
