Question

a student group sells 500 raffle tickets for $2 each. at the drawing the top prize will be a gift certificate for $100. second prize will be a $50 gift card and there will be five 3rd prizes, each a $20 gift card. do you expect to win or lose (on average)? how much? lose an average of $10.00 lose an average of $5.50 lose an average of $1.50 win an average of $2.00

Explanation:

Step1: Calculate the total revenue

The total revenue from selling 500 raffle tickets at $2 each is \(500\times2 = 1000\) dollars.

Step2: Calculate the total cost of prizes

The top - prize is $100, the second - prize is $50, and there are five third - prizes each of $20. So the total cost of prizes is \(100 + 50+5\times20=100 + 50 + 100=250\) dollars.

Step3: Calculate the expected value per ticket

The net profit is \(1000 - 250=750\) dollars. The expected value per ticket is \(\frac{750}{500}=1.5\) dollars. But since we are asked about the expected win or loss per ticket:
The formula for expected value \(E(X)\) is \(\sum_{i}x_ip_i\). Let \(x_1\) (winning the top - prize) have a probability \(p_1=\frac{1}{500}\), \(x_2\) (winning the second - prize) have a probability \(p_2 = \frac{1}{500}\), \(x_3\) (winning a third - prize) have a probability \(p_3=\frac{5}{500}\), and \(x_4\) (losing) have a probability \(p_4=\frac{500-(1 + 1+5)}{500}=\frac{493}{500}\)

\(E(X)=(100)\times\frac{1}{500}+(50)\times\frac{1}{500}+(20)\times\frac{5}{500}+(- 2)\times\frac{493}{500}\)

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Answer:

You expect to lose an average of \(\$1.50\)