Question

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

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

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: \(P(win) = \frac{1}{1,000,000}\)

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: \(Expected\ value\ of\ winning\ ticket = P(win) \times prize\ amount\) \(Expected\ value\ of\ winning\ ticket = \frac{1}{1,000,000} \times \$10,000 = \$0.01\)

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.

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.