What are the Chances? Student Handout
Playing the Lottery 1: What are the Chances?
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Students investigate their chances of winning the lottery using simple and compound probabilities (with replacement) represented by tables, tree diagrams, organized lists. From these, they develop algorithms for calculating compound probability using multiplication or exponents.
This lesson can be followed by What is the Cost?, where students compare prize amounts with the chances of winning those prizes and use compound probabilities (without replacement).
- Optional: Pick 3 Pennsylvania lottery card
- 10 index cards or slips of paper with the numbers 0-9 written on them or ping pong balls with the numbers 0-9 written on them in black marker
Students should have a basic introduction to probability before this lesson, including possible outcomes and representing probabilities as fractions and decimals.
Elicit students' knowledge and beliefs about the lottery by asking questions like the following:
- Has anyone here ever played the lottery? Do you know anyone who has?
- How does the lottery work?
- Do you think that playing the lottery is a good way to make money? Why or why not?
Clarify that a lottery is a game in which players pay small amounts of money for the chance of winning much larger amounts. The Pennsylvania Lottery offers prizes ranging from $1 or an extra card to millions of dollars.
- What do you think your chances are of winning?
- Do you think more people win nothing, a small prize ($1-$500) or the jackpot (millions)? Why?
End the discussion by explaining that today you will be using mathematics to calculate the chances or probability of winning some of the different games in the PA lottery.
- What information do you think we would need to have in order to do that? (Try to get to the idea that you need to figure out how many possible outcomes, or numbers, there are.)
Introduce Pick 3, a popular game in the PA lottery that is described on the What are the Chances? Student Handout. Ensure that students understand that three winning balls are drawn each day, and the same number can be drawn more than once. Simulate the lottery by having every student write down a 3 digit number, then pick randomly 3 times from the ping pong balls or the cards to determine the winning number. (The chances are quite slim that any of your students will "win" but some may come close)
How many combinations of 3 numbers do you think there are in total? Collect estimates at this point.
- What if we were just picking one number? What would our chances be of getting the right number? (1/10)
- Now lets think about what would happen if we picked two numbers.
There are several visual methods you can use to show the sample space and determine the number of possible outcomes: making an organized list, drawing a tree diagram and making a table. Begin to draw each of these and then stop to as:
- How many there will be and how do you know? (There are 10x10 or 100 possible outcomes)
Students should realize while making a table, tree, or ordered list that they do not need to write out every possibility in order to figure out how many there are. For example, just labeling the columns and rows of a table quickly demonstrates that there are 10 × 10 = 100 possible outcomes, and writing an organized list is the same as writing all of the 2-digit numbers from 00 to 99.
- Now what if we add a third number?
Allow students to make predictions but tell them that they will work together to figure that out in the exploration. Some may recognize that each of the each of the previous combinations of 2 digits now has 10 more possible combinations with the addition of the third digit or 10x10x10 = 1000. Other students might have misconceptions or not recognize this multiplicative structure yet. Use this as an opportunity to assess student thinking to see who will need more support in the exploration.
Review the three types of bets that can be made:
|
Description of Bet Types |
If the Winning Balls are 123, you would win if you got… |
|
STRAIGHT: Choose three digits. If all three are drawn in the same order, you win. |
123 exactly |
|
BOXED: Choose three digits. If all three are drawn in any order, you win. |
231 or any other order of the numbers 1, 2, and 3. |
|
FRONT PAIR: Choose two digits in the front. If both are drawn in the same order, you win. |
21* or 12* (The * means that the last ball doesn’t count.) |
|
BACK PAIR: Choose two digits in the back. If both are drawn in the same order, you win. |
*32 or *23 (The * means that the first ball doesn’t count.) |
Ask students to predict which bet leads to the best chances of winning.
- What do we need to know in order to figure this out?
Guide students to see that they need to consider how many possible combinations of numbers there are and also how many numbers will be considered a "win" for each of the games. Make sure they understand that straight has only one number that is considered a win while the other 3 have multiple combinations that can be considered a win.
The first task is for students to figure out how many possibilities there are when making a 3 digit number from 10 possible digits. If students are unable to get started after several attempts, suggest that they use the combinations that they already calculated from front pair to get started.
- To think about creating a table, one dimension should hold the all 100 pairs representing the first two balls (which we already calculated for front pair): 00, 01, 02…. 97, 98, 99. The 100 pairs can be combined with 10 choices for the last ball, to find 100 × 10 = 1,000 possible outcomes.
- In a tree diagram, students should observe that each of the 10 choices in the first level has 10 more choices in the second level to get 10 × 10 = 100 outcomes. Each of these has 10 more outcomes in the third level, so there are 100 × 10 = 1,000 possibilities.
- Using an organized list, students will likely notice that they’re writing all 3-digit numbers from 000 to 999, and that there are 1,000 of them.
The handout then begins with front and back pair, since those are easier and then moves to straignt and boxed.
Remind students that they can use tree diagrams, tables, or organized lists, and encourage each student or group to try the approaches that make the most sense to them. Once students begin, the will likely notice soon that there are more efficient ways to find the total number of possibilities by identifying patterns and rules after they write the first few stages of the organizational structures.
As students work, walk around to help them articulate their thinking mathematically as they find more efficient ways to solve the problems such as the following:
- Front Pair: You can consider the first two numbers only, so this is like finding the probability of picking a two digit number, or 1/100. Or, once you pick the first two numbers, you have 10 possible combinations that could win, or 10/1000. Either way the probability is 0.01.
- Playing Straight, page 5: Since they have already figured out the total number of possibilities is 1000, there is only one way to win, or 1/1000 or 0.001.
- Playing Boxed, page 6: There are still 1,000 possible outcomes, but we need to calculate the winning outcomes. It is important to be systematic and avoid counting the same order twice.
- If students make an organized list, notice if they write the numbers systematically or arbitrarily for discussion later.
- A tree diagram shows how the possibilities diminish at each level in an instructive way. By the last level, there is only one choice for each outcome.
- A table can be used with some creativity. Because the choice of the first and second ball leaves only choice for the third ball, the rows and columns can be used for the first and second ball only. Cells that would result in the same ball being picked twice must be crossed out.
Make sure all students are able to determine the sample space of 1000 and explore at least one of the options before moving to the whole class discussion.
For each of the bets, collect student responses for the number of possible wins. Guide students, as necessary, to understand that dividing the number of winning outcomes by the number of total outcomes results in the probability of winning, and that a probability close to 1 has a high chance of occurrence while a probability close to 0 has low chance.
Invite students to share their strategies on the board. This opportunity to share strategies is key to developing students as mathematicians. If any of the strategies listed above or in the Possible Student Solutions were not shared, present them to students as alternatives. In each strategy, highlight where multiplication is occuring.
Guide students to generalize a rule for calculating compound probabilities:
- Ask students if they can find a way to determine the number of possible outcomes for picking any number of balls any number of times without using tree diagrams, tables, or ordered lists. Work with students to generalize a rule based on multiplication:
(number of choices for the first digit) × (number of choices for the second digit) × … - Extend the example to illustrate the generalization: If there are the same number of choices for each drawing (10, in this example), then you can multiply that number by itself for each drawing. How many outcomes would there be if we drew 6 balls (with replacement) that were each labeled 0 to 9? (10 × 10 × 10 × 10 × 10 × 10 = 1,000,000)
- Guide students to recognize that this idea can also be written with exponents. There are 106 possible outcomes. The general rule can be expressed with variables: If there are c choices for each drawing and d drawings, there are cd possible outcomes.
Discuss with students how working with probability helped them think about playing the lottery by asking the same question that you asked at the beginning of the lesson:
- Do you think that playing the lottery is a good way to make money? Why or why not?
- If you changed your mind, what ideas changed your thinking?
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!”
In preparation for Playing the Lottery 2: What's the Cost?, have students make a quick calculation of the cost of playing the lottery:
- A lottery ticket costs $1. If you played once a day, how much would it cost you per year? How much would you have to win to make a profit?
The following websites may be useful for further exploring the lottery and compound probability:
- http://www.palottery.state.pa.us/
- http://www.palottery.state.pa.us/Games/PICK-3.aspx
- http://www.encyclopedia.com/topic/lottery.aspx
- https://www.mathsisfun.com/combinatorics/combinations-permutations-calculator.html
To continue exploring probability and the lottery, see Playing the Lottery 2: What's the Cost?
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