Question

one thousand chances are sold at $4 apiece for a raffle. there is a grand prize of $650, two second prizes of $200, and five third prizes of $50. first calculate the expected value of the lottery. determine whether the lottery is a fair game. if the game is not fair, determine a price for playing the game that would make it fair. the expected value of the game is $ - 2.7. (type an integer or a decimal. round to the nearest cent.) if the game is not fair, determine a price for playing the game that would make it fair (type an integer or a decimal. round to the nearest cent. type f if the game is fair.)

Explanation:

Step1: Calculate the probability of each outcome

The total number of chances \(n = 1000\).

• Probability of winning the grand - prize \(P(\text{grand})=\frac{1}{1000}\)
• Probability of winning a second - prize \(P(\text{second})=\frac{2}{1000}\)
• Probability of winning a third - prize \(P(\text{third})=\frac{5}{1000}\)
• Probability of losing \(P(\text{lose})=\frac{1000-(1 + 2+5)}{1000}=\frac{992}{1000}\)

Step2: Calculate the expected value formula \(E(X)=\sum_{i}x_{i}P(x_{i})\)

Let \(x_{1}\) be the net gain for the grand - prize winner (\(x_{1}=650 - 4=646\)), \(x_{2}\) be the net gain for a second - prize winner (\(x_{2}=200 - 4 = 196\)), \(x_{3}\) be the net gain for a third - prize winner (\(x_{3}=50 - 4=46\)), and \(x_{4}\) be the net gain for a loser (\(x_{4}=-4\))

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Step3: Determine the fair price

For a fair game, the expected value \(E(X) = 0\). Let the price of the ticket be \(p\).

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Answer:

The fair price for playing the game is \(\$1.30\)