Question

for the wheel pictured on the right, assume that a person spins the pointer and is awarded the amount indicated by the pointer. if it costs $10 to play the game, determine (a) the expectation of a person who plays the game and (b) the fair price to play the game.

a. what is the expectation? $6
(round to the nearest cent as needed.)
b. what is the fair price to play the game? $square$
(round to the nearest cent as needed.)

Explanation:

Step1: Calculate the probability of each outcome

The wheel is divided into 3 equal - sized parts. So the probability of landing on \( \$0\) is \(P(0)=\frac{1}{3}\), the probability of landing on \( \$10\) is \(P(10)=\frac{1}{3}\), and the probability of landing on \( \$20\) is \(P(20)=\frac{1}{3}\). But we need to consider the cost of playing the game (\(C = 10\)). The net - gain values are: \(x_1=0 - 10=- 10\), \(x_2=10 - 10 = 0\), \(x_3=20 - 10=10\).

Step2: Calculate the expectation

The formula for the expectation \(E(X)=\sum_{i = 1}^{n}x_ip_i\).

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Step3: Calculate the fair price

The fair price is the expected value of the amount won without subtracting the cost. Let \(y_1 = 0\), \(y_2=10\), \(y_3 = 20\). Using the formula \(E(Y)=\sum_{i = 1}^{n}y_ip_i\)

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Answer:

a. The expectation is \(\$0\).
b. The fair price to play the game is \(\$10\).