Everyone knows that winning the lottery is really, really unlikely. But sometimes those Powerball jackpots get really, really huge. So tempting! Is there a jackpot amount that makes the \$2 ticket worth the risk?

In this lesson, students compute the probability of winning the Powerball jackpot, and also the probabilities of other winning outcomes. Using the payouts for the outcomes, they find the expected value of a Powerball ticket. Finally, students decide whether the Powerball jackpot amount is ever large enough to justify buying a ticket.

### Students will

• Use combinations to calculate the probability of winning the Powerball jackpot
• Consider whether a large enough jackpot could guarantee you'd win money, if you bought all the tickets, given that multiple winners would split the winnings
• Consider the effect of probabilities and payouts on one's willingness to play Powerball
• Compute probabilities of other winning outcomes
• Use all of the probabilities and payouts to find the expected value of a Powerball ticket with a given jackpot
• Find the amount of the jackpot necessary to justify purchasing a ticket, based on its expected value alone
• Synthesize all the considerations in order to justify a player's willingness to play Powerball, or not

### Before you begin

Students should know how to compute probabilities that require use of combinations to count outcomes. This lesson serves as a context for applying that technique. Students should be able to find the expected value of a bet given the probabilities of winning and payoff amounts for the various outcomes. Expected value is used as a vehicle for making decisions in this lesson, but the idea should be developed beforehand.