Question

a certain game consists of rolling a single fair die. if a four or five comes up, you win 8 dollars, otherwise, you lose 4 dollars. find the expected winnings for this game. is this a fair game? the expected value for this game is dollars.

Explanation:

Step1: Calculate the probability of winning

A fair die has 6 possible outcomes. The numbers 4 and 5 are 2 winning outcomes. So the probability of winning \( P(\text{win})=\frac{2}{6}=\frac{1}{3}\).

Step2: Calculate the probability of losing

The probability of losing \( P(\text{lose}) = 1 - P(\text{win})=1-\frac{1}{3}=\frac{2}{3}\).

Step3: Use the expected - value formula

The expected - value formula is \(E(X)=x_1P(x_1)+x_2P(x_2)\), where \(x_1\) is the winning amount (\(x_1 = 8\)) and \(x_2\) is the losing amount (\(x_2=- 4\)).

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Answer:

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