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Chapter 9: Problem 538

The State of New Hampshire conducts an annual lottery to raise funds for the school districts in the state. Assume a million tickets are sold. One ticket is the winning ticket and the winner receives \(\$ 10,000\). If each ticket costs \(\$ .25\), find the expected value of a randomly purchased ticket and the revenue that the lottery generates for the school districts in the state.

Short Answer

Expert verified
The expected value of a randomly purchased ticket is -\$0.24, indicating an average loss of 24 cents per ticket for the buyer. The lottery generates \$240,000 in revenue for the school districts in the state.

Step by step solution

01

Calculate the probability of winning

First, let's find the probability of winning the lottery, which is the ratio of winning tickets to the total number of tickets: \(P(win) = \frac{number\ of\ winning\ tickets}{total\ number\ of\ tickets}\) There is only one winning ticket among a million sold tickets, so the probability of winning is: \(P(win) = \frac{1}{1,000,000}\)
02

Determine the expected value of a winning ticket

To find the expected value of a winning ticket, we should multiply the probability of winning by the prize amount: \(Expected\ value\ of\ winning\ ticket = P(win) \times prize\ amount\) \(Expected\ value\ of\ winning\ ticket = \frac{1}{1,000,000} \times \$10,000 = \$0.01\)
03

Calculate the net expected value of a ticket

Now we need to subtract the ticket cost from the expected value of a winning ticket to get the net expected value: \(Net\ expected\ value\ of\ a\ ticket = Expected\ value\ of\ winning\ ticket - cost\ of\ a\ ticket\) \(Net\ expected\ value\ of\ a\ ticket = \$0.01 - \$0.25 = -\$0.24\) So the expected value of a randomly purchased ticket is -\$0.24, which means that on average, a ticket buyer will lose 24 cents per ticket purchased.
04

Calculate the revenue generated for school districts

The total revenue generated by ticket sales is given by: \(Total\ revenue = number\ of\ tickets \times cost\ per\ ticket\) \(Total\ revenue = 1,000,000 \times \$0.25 = \$250,000\) Now, let's find the revenue generated for the school districts by subtracting the prize money from the total revenue: \(Revenue\ for\ districts = Total\ revenue - prize\ amount\) \(Revenue\ for\ districts = \$250,000 - \$10,000 = \$240,000\) The lottery generates \$240,000 in revenue for the school districts in the state.

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