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Community Based Mathematics Project of Philadelphia

Playing the Lottery 2: What's the Cost?
Submitted By: CBMP

    Display Image
    Graphic of Pennsylvania Lottery logos on several lottery balls.
    Grade
    Grade
    7th Grade
    Math Focus
    Math Focus
    Compound Probabilities
    Context
    Context
    Everyday Life
    Sports & Games
    Common Core State Standards
    Statistics & Probability
    Investigate chance processes and develop, use, and evaluate probability models. : Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. (Math.7.SP.C.8)Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. (Math.7.SP.C.8a)Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., "rolling double sixes"), identify the outcomes in the sample space which compose the event. (Math.7.SP.C.8b)
    Overview / Lesson Summary

    Students build on the strategies they developed in Playing the Lottery 1: What are the Chances? which introduces easier lottery games using simple probability and compound probability with replacement.

    In this lesson, students calculate the probability of winning the lottery in a game and explore factorials. They use probability, estimations, and percents to understand the cost of playing the lottery and who really makes money.

    Preparation and Materials
    • Optional: Pick 3 and Cash4Life Pennsylvania lottery cards to share as a model.
    Introducing the Context

    Show the following chart (which is also on the Student Handout). If you have taught Playing the Lottery 1: What are the Chances? review the questions and conclusions that were generated about Pick 3. If not, use the chart to make some conjectures.

    Pick 3 Bet Types

    For every $1 you bet, you win…

    Straight (all 3, same order)

    $500

    Boxed (all 3, any order)

    $80

    Front pair (just two, same order)

    $50

    Back pair (just two, same order)

    $50

     

    • Why do you think the payouts are different for each game?
    • How do you think they determine these amounts?
    • What are your chances are of winning any of these games? Why?

    Possible responses (based on calculations from Playing the Lottery 1: What are the Chances?) include:

    • There is a pretty good chance (1/50) of winning front pair, so many people will probably win it and the pay-off isn’t very high. This is the same chance as winning back pair, so the pay-off is the same.
    • There is only a 1/1,000 chance of winning while playing straight, so few people will win it and the pay-off is very high.
    • The chance of winning while playing boxed is in between (6/1,000) so the winning amount is in between.

    Have students share additional questions or observations. 

    Ask students to share their opinions on a few questions, such as:

    • Do you think that playing the lottery is a good way to make money? Why or why not?
    • Do you think more people win nothing, a small prize ($1-$500), or the jackpot (millions)?

    Key contextual vocabulary: lottery, winnings, jackpot, odds, probability

    Introducing the Mathematical Ideas

    Review the mathematical rule for calculating possible outcomes that they found during Playing the Lottery 1: What are the Chances?

    To find the number of possible outcomes (when order is important), use the number of choices for each ball as the number to multiply, and the number of balls for the number of times to multiply it. For example, if there are 3 balls and each can have the numbers 0-9, then there are 10 × 10 × 10 or 103 possible outcomes. Work with students to generalize this and review how this rule is derived using tables, trees, and organized lists. 

    • If there are c choices for each drawing and d drawings, there are cd possible outcomes.

    Explain to students that they will use the Exploration to investigate three questions:

    • How are prizes calculated?
    • Can you guarantee a win?
    • What affects your chances of winning different games?

    Ask students what they could do to guarantee that they get a winning number. They should come to the conclusion that the only way to guarantee a win would need to buy tickets with every possible combination of the 3 numbers. Explain that first they will think about these questions for Pick 3, and then for a more complex game called Cash4Life. Show students the Pick 3 and Cash4Life tickets if you have them.

    Key mathematical vocabulary: odds, probability, compound probability, exponent, outcomes, table, tree, organized list

    Exploration

    Students should be able to extend and draw on their work from What are the Chances? when working on the explorations. When they get to Cash4Life, they will need to develop some new ways of solving the problems because the game is more complex.

    Assign students to work independently or in small groups, then circulate to support them. The last question is more difficult.

    • Pages 3 and 4: The challenge problem for Pick 3 Boxed (calculating probabilities where order is not important) builds on the strategies that students will explore in two easier problems that students will explore in Cash4Life on page 6. If students are struggling with these challenge problems, have them continue on and then return later.
    • Pages 6 and 7: Students may initially assume that they can calculate the outcomes on these pages using the same strategies that they used in lesson 1 when the numbers could be re-used. If this occurs, point out that this strategy would generate the numbers 00000, 00001, etc. which are not permitted because the numbers repeat. Encourage students to draw a tree and make sure not to include choices which are not available. The problem on page 5 of the Lesson 1 student handout (where students calculate the chances of winning Pick 3 Boxed) shows the same concept with even fewer choices, and may be useful as a reference.
    Discussion

    Mathematical Discoveries

    Help students formalize the strategies that they used to calculate the number of outcomes when balls are selected without replacement and when order is not important using page 8 of the student handout. See the Possible Student Solutions for ideas. 

    Guide students to articulate the following ideas:

    • When you have to pick numbers without repeating, the chance for the first ball is the total number of options, and the chance for the next ball is one less than that because one has already been used up. The next place has two fewer options because two have been used up. Each place has one fewer choices left.
    • As in lesson 1, we can calculate the total number of “leaves” or outcomes on the tree by multiplying the number of options in each level together. For example, to arrange 5 balls, there are 5 × 4 × 3 × 2 × 1 = 120 ways to do it.
    • This can also be written with variables. If there are n choices, there are n × (n – 1) × (n – 2) × (n – 3) × …, possible outcomes.

    Introduce factorials which are represented by an exclamation mark. Thus, n × (n – 1) × (n – 2) × (n – 3) × … × 1 = n! or “n factorial.” Explain that factorials are a product of all of the numbers from 1 to the number of choices.

    Optional: You may want to explain how dividing a factorial by a factorial can be used to get only the first part of a factorial, because the factors cancel out. For example: 75 × 74 × 73 × 72 × 71 = 75!/70! See the Possible Student Solutions for more information.

    Discuss that when the order is not important, you can divide the total number of outcomes by the number of ways that the digits can be rearranged. 

    Ideas about Playing the Lottery

    Discuss with students whether they think playing the lottery is a good idea by reexamining some of the questions from the class list, by revisiting the questions that guided the lesson, or with some of the other questions below:

    • How are prizes calculated? (pages 2 & 6) Does understanding the relationship between the amount of money you could win and your chance of winning change how you think of the lottery?
    • Who makes money when you play the lottery? (pages 3 and 5) Does calculating how much money the state keeps change how you think about playing the lottery?
    • Can you guarantee a win? (pages 3 and 4) After calculating the amount of money it would take to guarantee winning, have your feelings about playing the lottery changed? Do you think that it is worth it?
    • Over time, do you think the majority of people win or lose money playing the lottery? Why?
    • Does this match with your experience? Think of the people you know. How often do they play the lottery? About how much do they spend each day/week/month? How much have they won?
    • Do you think that there is a better way for people to spend their money? Do you think there is a better way for them to make more profit?
    Practice

    If students were unable to solve the Challenge Problems on pages 3 and 4 of the Student Handout, you may want to assign them for homework to reinforce their understanding of the rule they discovered today and provide an additional challenge. 

    Extensions

    Read or give students the statements below about picking numbers for the lottery. Have them decide whether they agree or disagree using probability.

    • “I play the same numbers every day. They’re bound to win soon!”
    • “These numbers have never won. It’s their turn!”
    • “These numbers have won more than any others. They will probably win again!”
    • “These are my lucky numbers! They will definitely win!”

    Read more about the lottery with these Websites:

    To continue exploring probability and the lottery, complete Lesson 3.

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