Document 6484450

 © Mathalicious 2014 lesson guide ODDSBALLS When is it worth buying a Powerball ticket? 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. Primary Objectives 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 •
•
•
•
•
•
•
Content Standards (CCSS) Statistics CP.9, MD.5, MD.7 Mathematical Practices (CCMP) Materials MP.1, MP.2, MP.3 •
•
•
Student handout LCD projector Computer speakers Before Beginning… 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. Lesson Guide: ODDSBALLS 2 Preview & Guiding Questions Since students may be too young to legally play the lottery, we'll start by familiarizing them with the Powerball game. Tell students that to play Powerball, you choose five different "white" numbers between 1 and 59. You also choose one "red" number (The Powerball!) between 1 and 35. Ask students to create their Powerball ticket by jotting down their play: five white numbers and one red number. If all six of those numbers matched the drawing, they would have won the jackpot on that day! Then, play the video of the Powerball drawing. (We provided three different drawing videos, to prevent students from tipping each other off if you teach more than one section. Feel free to repeat the drawing with a different video, if your students would benefit from watching more than one.) Once students have a basic familiarity with the Powerball game, they're ready to roll into the first act. •
•
•
•
•
•
Does anyone have a family member who has played Powerball, or another state lottery? Has anyone watched a lottery drawing on TV? What numbers can the player choose from to create their lottery ticket? Can a player choose the same white number more than once on the same ticket? Can a ticket's red number be the same as any of the ticket's white numbers? Does anyone's ticket match all six numbers in the drawing? Act One To start off, students compute the probability that a Powerball ticket matches all six numbers. This is a matter of counting all the possible ways to choose five numbers out of 59 and the ways to choose one number out of 35. The probability of matching all six numbers and winning the grand prize is, of course, vanishingly small. Next, students consider whether the size of the jackpot affects their willingness to play. If the jackpot were large enough, could they simply buy all possible tickets and guarantee themselves a win? Nope! This is because if there is more than one winning ticket, the prize is split. Matching all six numbers does not guarantee you will win the entire jackpot; in fact, you may only win a fraction of the jackpot. Finally, students learn that there are more ways to win at Powerball than just by matching all six numbers. To wrap up the first act, students start to grapple with the nuances of computing the probabilities of obtaining different results, and also start to consider whether the payout for a particular outcome affects their willingness to play. Act Two In Act Two, students develop – with much more precision – the probabilities of winning with different outcomes. They also quantify the effect of the payout on their willingness to play by computing the expected value of a Powerball ticket. They consider whether there could be a jackpot large enough to make them willing to play. Finally, students see that a huge jackpot also means a huge number of players buy tickets. The larger the number of tickets sold, the more likely it is that the jackpot will get split. In the end, students will understand how all the different variables might affect the decision to play, and that some Powerball drawings are a better bet than others. Lesson Guide: ODDSBALLS 3 Act One: Let's Play Powerball! 1 Powerball is one of the most popular lotteries in the United States. Six balls are drawn at random: five white balls (numbered 1-­‐59), and the red Powerball (1-­‐35). Once a ball is drawn, it is not replaced. To win the grand prize, a player must correctly pick all six numbers. Note: when a player buys a ticket, the white numbers are automatically printed in ascending order. In the example below, the ticket 2-­‐11-­‐19-­‐27-­‐56-­‐17 would win. If you purchased a single Powerball ticket, what is the probability that you would win the grand prize? White (1-­‐59) 19 11 27 56 2 Powerball (1-­‐35) 17 There are 35 possible outcomes for the red ball. Since the order doesn't matter, and white balls are drawn 59
without replacement, there are = 5,006,386 possible outcomes for the white balls. 5
Since choosing one red ball and choosing five white balls are independent events, there are 5,006,386 × 35 = 175,223,510 unique outcomes in the Powerball drawing. Therefore, the probability of winning the grand prize is 1
175,223,510
or approximately 5.7 × 10-­‐9. Explanation & Guiding Questions Depending on students' experience with combinations vs. permutations, they may think there are 59 × 58 × 57 × 56 × 55 = 600,766,320 possible outcomes for drawing five white balls. That would be correct if 19-­‐11-­‐27-­‐56-­‐2-­‐17 were a different ticket from 2-­‐11-­‐19-­‐27-­‐56-­‐17; however, all tickets are automatically printed in ascending order. Therefore, these two examples are the same outcome. It may be instructive to have students explore how they could modify a calculation of 600,766,320 to get rid of the redundant outcomes. Each unique outcome within the 600,766,320 is counted 5! times (the number of ways to arrange five things), so they'd divide the result by 5! or 120. The 5,006,386 outcomes for white balls can only be multiplied by the 35 outcomes for red balls because they are independent events. You may choose to emphasize this, or not, depending on your learning goals for the class. Once students correctly ascertain that there are 175,223,510 possible tickets, they must then reason that only one of those outcomes corresponds to the winning ticket. Therefore, the probability (proportion of total outcomes that 1
are successful) that the single purchased ticket would win the grand prize would be . 175,223,510
•
•
•
•
•
•
•
How many white balls are there to choose from? How many red balls? Describe the process of choosing the five white balls. Would drawing 19-­‐11-­‐27-­‐56-­‐2-­‐17, compared to 2-­‐11-­‐19-­‐27-­‐56-­‐17 result in different tickets? Why would computing 59 × 58 × 57 × 56 × 55 give us a result for white balls that was too large? Does the red ball that is drawn depend in any way on the white balls that are drawn? If you know the probabilities of independent events, how can you find the probability they occur together? Once we know the possible number of tickets, how can we find the probability that one ticket is a winner? Deeper Understanding •
How many outcomes would there be if the white balls were replaced between drawings, such that numbers could appear more than once on a ticket? (595 × 35 = 25,022,350,465.) Lesson Guide: ODDSBALLS 4 2 The minimum grand prize is $40 million. While there is no maximum, the largest ever was close to $591 million. Tickets cost $2 each. If the jackpot were $400 million and you could somehow buy as many tickets as you wanted, do you think it would be worth playing? Explain your reasoning. Answers will vary. Trying to guarantee a win would require spending over $350 million to purchase all the possible tickets. If the jackpot were greater than $350 million dollars (and someone had that much to spend on tickets) this strategy might make sense. However, if someone else happens to also hold a winning ticket, the winners would split the pot. Perhaps $350 million is not a high enough jackpot for this strategy to work. Explanation & Guiding Questions Students should be encouraged to use the insight gained in the previous question to inform their decision. Some may only consider buying one ticket, and reason (as millions do) that the probability of winning is so vanishingly small that no jackpot amount would be worth it. Others may reason (like the millions who play Powerball every week) that even a tiny chance is still a chance, and $2 is not much to lose. If students grasp that buying more than one ticket will increase the probability of winning, they may wonder how buying more than one ticket will change the probability of winning. For relatively small numbers of tickets, students can calculate that the effect on the probability of winning would be negligible. For example, even if someone spent $2000 on 1000 tickets, the probability of winning would go from an order of 10-­‐9 to 10-­‐6. Better, sure, but a 0.00057% chance is still awfully close to impossible. Some students may extend this to a strategy of buying all the tickets, leading to the line of reasoning in the sample response. Seems like a great plan, but the fact that multiple winners split the pot really throws a wrench into the works, since there is no guarantee that buying up all the numbers would result in winning the entire jackpot. •
•
•
•
•
•
What is the probability of winning with one ticket? Why do you think many people play Powerball, even though their chance of winning is so very small? What could a player do to increase his chance of winning the jackpot? How would the probability of winning change if a player bought 1000 tickets? How much would he spend? How much would you have to spend to buy all the possible tickets? Does buying all of the tickets guarantee you will win the jackpot amount? Why or why not? Deeper Understanding •
•
If you were the sole winner of a $350 million jackpot, would you actually receive $350 million? (No. Big lottery winnings are taxed to the tune of 50%.) Are there any strategies that would guarantee a player would win money at Powerball? (Nope! The only way to guarantee a winning ticket is to buy all possible tickets. However, there is always a possibility that the jackpot will be split many ways. Since the actual amount you might win is unknown, there is no jackpot amount that makes it safe to assume you'll come out ahead.) Lesson Guide: ODDSBALLS 5 3 The likelihood of winning the jackpot is very low. However, there are other prizes, too. For instance, if a player matches the red ball only, he wins $4. According to the Powerball website, the probability of this is 1 in 55.41. a. Why isn’t the probability of this outcome 1 in 35? b. Do you think $4 is a good payout? Explain. There chance of matching the red ball only is less than There's a 1 in 55.41, or 1.8% chance of doubling your 1 in 35. To get this outcome, you'd have to money. Opinions will vary as to whether that payout is simultaneously NOT match any of the white balls. The a good one. "red ball only" outcome is really "match the red ball and also zero white balls." Therefore, the probability is less than 1/35. Explanation & Guiding Questions In order to explain why the probability of matching the red ball only is not 1 in 35, students must consider what is going on with the white balls. They have already seen a case where matching the red ball was part of a different prize: the grand prize from matching the red ball and also all six white balls. Since "red ball only" means "match the red ball and also zero white balls," students should be able to reason that this is less likely than the 1/35 chance of matching the red ball without regard to white balls. There's a 1 in 35 chance that we match the red ball and also all the white balls, or four of the white balls, or three of the white balls, etc. When students consider whether $4 is a good payout, they should reason that it makes sense for outcomes with a higher probability of occurring to have a lower payout. Whether they actually think a $4 payout is "good" is irrelevant; the important idea is that when the probability of winning is lower, the player needs the payoff to be higher to feel as if it's a good bet. •
•
•
•
•
If you match the red ball, what other prize could you win, besides "red ball only?" If a ticket only matches the red ball, how many white balls does it match? Why is "red ball only" less likely than 1 in 35? Why do you think the payout for matching the red ball only is so much lower than the grand prize? Do you think $4 is too low? Or do you think the higher probability of winning this prize justifies it? Deeper Understanding •
54
ways to 5
choose 5 whites that do not match any of the 5 whites. Number of ways to match the red ball and no white balls is 3,162,510. There are still 175,223,510 possible outcomes. 175,223,510/3,162,510 ≈ 55.41.) How would you compute the 1/55.41 probability? (There is one way to match the red and Lesson Guide: ODDSBALLS 6 4 There are nine different ways to win Powerball. Choose one of the outcomes, and calculate its probability. (Then collect the remaining probabilities from your classmates.) Match Payout P $40 million Ways to choose matching whites Ways to choose non-­‐
matching whites Ways to choose matching reds Ways to choose non-­‐
matching reds 1 1 1 1 1 in 175,223,510 ≈ 5.8 × 10-­‐9 34 34 in 175,223,510 ≈ 1 in 5,153,633 ≈ 1.9 × 10-­‐7 1 270 in 175,223,510 ≈ 1 in 648,976 ≈ 1.5 × 10-­‐6 34 9180 in 175,223,510 ≈ 1 in 19,088 ≈ 0.000052 1 14310 in 175,223,510 ≈ 1 in 12,245 ≈ 0.000082 34 486,540 in 175,223,510 ≈ 1 in 360 ≈ 0.002777 1 248,040 in 175,223,510 ≈ 1 in 706 ≈ 0.001416 1 1,581,255 in 175,223,510 ≈ 1 in 111 ≈ 0.009024 1 3,162,510 in 175,223,510 ≈ 1 in 55 ≈ 0.018048 total outcomes = 1 1 P P P P P $1 million $10,000 1 1 total outcomes = 34 5
4
54 1 total outcomes = 270 $100 5
4
54 1 total outcomes = 9180 $100 5
3
$7 5
3
54
2
1 total outcomes = 14310 54
2
1 total outcomes = 486,540 $7 5
2
54
3
1 total outcomes = 248,040 $4 5 54
4
1 total outcomes = 1,581,255 $4 1 54
5
1 total outcomes = 3,162,510 Probability Lesson Guide: ODDSBALLS 7 Explanation & Guiding Questions for Question 4 The first row, the grand prize, was calculated previously, but is provided here for reference. The last row, matching the red ball only, had the result given previously, but wasn't calculated. Students who need a simpler problem should start with the second row: matching 5 white balls but not the red ball. Then, the last row: matching only the red ball (even though this result was given previously). Encourage students who are up for a challenge to try the computations that require more than one combination. (Note that the question only asks students to compute one of the probabilities.) The answers above are expressed as "1 in something" because the value has more meaning that way. Students may elect to compute decimal values if that is the way they normally express probabilities. But these alternate ways of expressing the answers are optional. The most precise form to use in the following question will be the number of outcomes out of 175,223,510. You can expect some students to think of the probability for the white balls separately from the probability for the red ball, and then multiply each. This is fine, since drawing white and drawing red balls are independent events. For example, the computation for matching all five white balls but not the red ball might look like: 1
34
1
34
× =
× 59 35 5,006,386 35
5
•
•
•
•
•
•
•
How many ways could you choose the matching (or non-­‐matching) red ball? How many ways could you choose the non-­‐matching white ball(s)? How many ways could you choose the matching white ball(s)? How many total outcomes were there? How can we use all this information to find the probability of one of these outcomes? The probabilities are easier to compare as "1 in something." How could we express them this way? How could we express them as a decimal value? Deeper Understanding •
For cases where there is only one way for an outcome to occur, can you express that as a combination? (As n
n
a combination, these would be expressed as either , when choosing none of the things, or , when 0
n
5
5
choosing all available things. For example, there are ways to choose no matching white balls, and 0
5
ways to choose all five matching white balls. Lesson Guide: ODDSBALLS 8 5 Determine the expected value of a Powerball ticket when the jackpot is $40 million. Based on this, do you think it’s worth buying a Powerball ticket? If not, what would the jackpot need to be to justify buying a ticket? (Note: if there are multiple jackpot winners, they split the award.) Match Payout Probability Payout × Probability $40 million 1 in 175,223,510 ≈ 0.2283 $1 million 34 in 175,223,510 ≈ 0.1940 $10,000 270 in 175,223,510 ≈ 0.0154 P $100 9180 in 175,223,510 ≈ 0.0052 P $100 14310 in 175,223,510 ≈ 0.0082 $7 486,540 in 175,223,510 ≈ 0.0194 P $7 248,040 in 175,223,510 ≈ 0.0099 P $4 1,581,255 in 175,223,510 ≈ 0.0361 $4 3,162,510 in 175,223,510 ≈ 0.0722 P P The expected value of a Powerball ticket is the sum of the "Payout × Probability" of each outcome: 0.5587, or about 56 cents. So when the jackpot is $40 million, a player can expect to lose $2 -­‐ $0.56 = $1.44. In order to break even, the expected value would have to be exactly $2. Let J represent the jackpot amount required to break even. (Payout × Probability of Jackpot) + (Payout × Probability of All other Outcomes) = $2 J × 1
175,223,510
+ 0.3604 = $2 J = $287,296,467 Explanation & Guiding Questions Once we're looking for the expected value of a Powerball ticket, the purpose of the Payout × Probability column is clear: the expected value of a ticket is just the sum of these products for each outcome. The most precise way to compute Payout × Probability is to use the exact probabilities (out of 175,223,510), since no rounding takes place before rounding the resulting product. With a $40 million jackpot, a player expects to win $0.56. A ticket would be "justified" if this expected value were greater than the $2 ticket price. If students consider the work they just did, they should notice that they'll simply need to write and solve an equation. The jackpot amount is unknown, but they want the result to be $2. •
•
•
•
•
How is expected value calculated? What is the most precise way to compute the Payout × Probability for each outcome? How can these be used to find the expected value of a Powerball ticket with a $40 million jackpot? What would the expected value need to be in order for a player to expect to break even? How could we write the expected value computation as an equation with an unknown jackpot amount? Deeper Understanding •
When the expected value is higher, does that mean the player is more likely to win? (No. The probability of winning does not change. Expected value just helps quantify whether a reward is worth the risk.) Lesson Guide: ODDSBALLS 9 6 Look at the data for every Powerball drawing in 2013: the date, jackpot, and number of tickets sold. Were there any drawings for which you would have wanted to buy a ticket? Explain your reasoning. It would make the most sense to buy a Powerball ticket if the jackpot were greater than around $287 million, as calculated previously. Looking at the table, there were six such jackpots. For example, on May 15, 2013, the jackpot was $360 million, and on May 18, 2013, it rose to $550.5 million. However, by looking at the graph, we can easily compare the jackpot with the number of tickets sold at each drawing. It is clear that when the jackpot gets large, many more people play. A much larger number of players would increase the chance of more than one winner, meaning the pot would be split. Because of this, perhaps it is not a good idea to play Powerball even when the jackpot is large. Explanation & Guiding Questions If students agree that playing Powerball is justified when the expected value of the ticket exceeds $2, it should be straightforward (based on previous work) to conclude that they would want to buy a ticket when the jackpot exceeds $287 million. Of course, this is not an automatic conclusion, as some students may still view a bet with such low probability of winning as a waste of money, whereas some students may still want to play despite the low probability of winning. This is a good opportunity to revisit the meaning of expected value: it's a way to quantify whether a risky bet is worth it, but it does not change the probability of winning or losing. If the expected value is more than $2, it doesn't mean the player should assume they will win more than $2. Some students may be satisfied with the conclusion that playing Powerball is justified whenever the jackpot exceeds $287 million. Encourage them to also consider the number of tickets sold. By directing their attention to the graph, they will see that a really large jackpot means a large number of tickets will be sold. A large number of tickets means a higher probability that there will be more than one winner, and the winners will have to split the pot. If only a fraction of the jackpot is won, the expected value calculation is kind of out the window. As a result, students may conclude that they'd only buy a ticket when the expected value reached $4. Or they may conclude that the higher probability of having to split the pot means that playing Powerball is never justified. Some students may be aware that winners do not take home the entirety of the pot. If the winner takes a lump-­‐sum payment, they receive about 2/3 of the jackpot amount. Additionally, lottery winnings are heavily taxed. Perhaps the actual payout is a better number to consider than the jackpot. Should it come up, it is certainly worth talking about. •
•
•
•
•
•
For what jackpot did the expected value reach $2? Has the Powerball jackpot ever exceeded that amount? When? If you were old enough, would you really have wanted to play at these times? Why or why not? At what times are a large number of tickets sold? How does a large number of tickets being sold affect the potential outcome of Powerball? Does a large number of tickets being sold affect your desire to play Powerball? Why or why not? Deeper Understanding •
The lottery typically gives you the option of taking the jackpot in equal payments spread out over 30 years, or a one-­‐time payment worth about 2/3 of the jackpot’s value. Which would you prefer and why? (Answers will vary, but in many cases investing the lump-­‐sum payment with compounding interest would more than make up the difference between the one-­‐time payment and the 30-­‐year annuity. Of course you’re going to lose a lot of that money to taxes either way.)