math statistics

Question

The student council is hosting a drawing to raise money for scholarships. They are selling tickets for $9 each and will sell 500 tickets. There is one $1,000 grand prize, four $400 second prizes, and sixteen $10 third prizes. You just bought a ticket. Find the expected value for your profit. Round to the nearest cent.

Answer 1

One $1,000 prize

Four $400 prizes

Sixteen $10 prizes

500 tickets

you bought one ticket costing $9

your expected winnings are a weighted average of the possible outcomes

1000/500 +4(400)/500 +(16)10/500 + (479)(0)/500 minus $9

= $2 + $3.20 + $0.32 - $9

= $5.52 - $9

= an expected loss of exactly $3.48

You won't actually ever win $3.48 with one ticket, but it's the expected average loss over time, if you play enough. Most times you play you'll just be out $9. That's sort of your realistic expected loss when playing one time. But not the answer they want for a math course.

If you bought all 500 tickets, you'd win 21 prizes worth $2,760 but pay 500 x $9 = $4,500

2760-4500 = -$1740

1740/500 = a loss of exactly $3.48 per ticket purchased = the mathematical expected loss

the student council will raise 3.48 x 500 = $1,740 for scholarships

if it' a fair drawing. But if someone cheats, your expected loss is greater

the student council will raise the same amount regardless if anyone cheats. Best to disqualify any student council members.