Name of Unit: Powerball

Subject and Grade Level: Math, Middle School

Objectives:

  • Identify whether an experiment is independent or dependent.
  • By using a simulation, understand the law of large numbers.

Time: 1 day

Minnesota State Standards Addressed:

Bold type indicates standards addressed by this unit.

A student shall:

  1. Evaluate and solve problems, including calculating basic measures of center and variability, to demonstrate understanding of basic concepts of probabilityand calculate simple probabilities;
  2. Formulate a question and design an appropriate data investigation;
  3. Organize raw data and represent it in more than one way;
  4. Analyze data by selecting and applying appropriate data measurement concepts;
  5. Critique various representations of data;
  6. Devise and conduct a simulated probability situation; and
  7. Predict future results based on experimental results.

Knowledge:

Identify independent events.

State the law of large numbers.

Skills:

Calculate simple probability for an event.

Run a simulation of a Powerball game.

Apply the law of large numbers.

Understanding:

Understand that the chances of winning the jackpot is almost the same as not playing at all.

Differentiate between independent and dependent events.

Lesson Plan

Prerequisite knowledge:

  • calculate simple probabilities
  • know difference between experimental and theoretical probabilities
  • convert from fractions to decimals

Prerequisite: TI-83 calculators or computer set up with Powerball program

  1. “How many of you would play a game where you would lose 97.1% of the time?” “Do you know any games that have this?” (Powerball)
  2. In small groups discuss and record on worksheet:

Does it matter whether I put the cube back in or not after the first draw for the second to happen? Will the probability be different? Why or why not?

--Ten cubes in a bag, 5 red and 5 blue. What is the probability of drawing a red on the first draw? (5/10 = ½) If you replace the cube, what is the probability of drawing a red on the second draw? (still 5/10 = ½)

--Same bag of cubes. What is the probability of drawing a red on the first draw? (5/10 = ½) If you DO NOT replace the cube, what is the probability of drawing a red on the second draw? (if the first cube is red, 4/9, if not, 5/9)

If you flip a coin 4 times and it comes up heads all four times, what is the probability the fifth flip will be heads? (1/2)

  1. Wrap up discussion as a whole class. Discuss dependent and independent events. Brainstorm examples of each.
  2. Lead into discussion of Powerball. What is it? How is it played? Is this an independent or dependent game?
  3. Run the Powerball simulation.
  4. Calculate experimental probabilities.
  5. Compare these to theoretical probabilities (given).
  6. Talk about the law of large numbers—graphic representation.
  7. Record class data on worksheet to complete as homework.

Assessment:

Worksheet (includes both in-class and homework portions)

Additional Resources

Lottery sales by capita:

Lottery history:

Powerball:

Worksheet

In-class:

  1. Suppose we have ten cubes in a bag, 5 red and 5 blue.

What is the probability of drawing a red on the first draw?

If you replace the cube, what is the probability of drawing a red on the second draw?

  1. Using the same bag of cubes.

What is the probability of drawing a red on the first draw?

If you DO NOT replace the cube, what is the probability of drawing a red on the second draw?

  1. If you flip a coin 4 times and it comes up heads all four times, what is the probability the fifth flip will be heads?
  1. Record the list of examples of independent and dependent events.
  1. Run the Powerball simulation. Record your results in the “Individual Probabilities” column. Record the “Class Probabilities”.

Individual Probabilities / Class Probabilities / Theoretical Probabilities / Which is closer to the theoretical probabilities—individual or class probabilities?
Jackpot / 0.00000001
$100,000 / 0.00000051
$5,000 / 0.00000275
$100 / 0.00023074
$7 / 0.00649651
$4 / 0.00847500
$3 / 0.01355999
Loss / 0.97123449

Homework:

  1. Finish the table above by comparing individual and class probabilities in the last column.
  1. Explain how the law of large numbers is shown in the table above.
  1. Identify whether the following events are independent or dependent.
  • Getting gumballs out of a gumball machine.
  • Flipping coin many times.
  • Rolling die many times.
  • Drawing straws.
  • Getting a full glass of milk out of the carton in the fridge.
  1. What makes events independent?
  1. What makes events dependent?