Summary
In this lesson, students will calculate the probability of a discrete random variable and represent the probability distribution as a table and histogram. Then, they will determine the mean and standard deviation using the probability distribution.
Essential Question(s)
How do I determine the likelihood of the outcomes of a scenario?
Snapshot
Engage 1
Students play the dice game “Greedy Pig” to investigate the best chance of winning the game.
Explore
Students imagine flipping a coin three times. Then, they record the possible outcomes and likelihood of the discrete random variable representing the number of heads.
Explain
Students complete guided notes as a group to learn more about discrete random variables, probability distribution, mean, and standard deviation.
Engage 2
Students challenge their knowledge of probability through the “Monty Hall Problem.”
Extend
Students watch an ICAP video and answer reflection questions about the quality control underwriter’s career.
Evaluate
Students complete a Choice Board to demonstrate their understanding.
Materials
Lesson Slides (attached)
Greedy Pig handout (attached; one per student; printed front only)
Heads or Tails handout (attached; one per pair; printed front only)
Benefits handout (attached; one per student; printed front/back)
Benefits (Model Notes) document (attached; one per student; printed front/back)
Choice Board handout (attached, one per student; printed front only)
Choice Board (Sample Responses) document (attached; for teacher use)
Video Reflection Questions handout (optional; attached; one per student; printed front only)
Sticky notes (optional)
Dice (one per student or one per pair)
Coins (optional; one per pair)
Pencils
Paper
Graphing calculators
Student devices with internet access (optional)
Engage 1
15 Minute(s)
Introduce the lesson using the attached Lesson Slides. Slides 3–4 display the lesson’s essential question and learning objectives. Review each of these with your class to the extent you feel necessary.
Give each student a copy of the attached Greedy Pig handout. Using slide 5, introduce the Greedy Pig game and explain the game’s rules and procedures. Explain to students that they will each roll a die. If they roll a 1, 3, 4, 5, or 6, they get to write down that value as points they earned. However, if they roll a 2, they lose all points for that round (they need to roll again if their first round rolls a 2).
After the first round, they get to decide if they want to stop and keep their points for the round, or roll again with the risk of losing their points. If they choose to roll again and get any number but 2, they get to add more points, but if they roll a 2, they lose all points for that round.
Pairs of students begin each round together but end each round independently. In other words, Students A and B both begin each round at the same time, but each decides for themselves when to end their round.
Display slide 6 to show students two round outcome examples. Once students understand the rules, go back and display slide 5 for their reference as they play. Have students work with their Elbow Partners to play the first five rounds and record their results on their handouts. After a few minutes, move to slide 7 and have the class answer:
How did you decide when to end your round?
What are some possible strategies to maximize your score?
After discussing strategies, have students return to their Elbow Partners and start a new five-round game.
After a few minutes, gather the whole class again and show slide 8. Determine two students with the highest scores during the second game. Then, have the two highest scorers share their scores and their strategies with the class. If both students have the same strategy, discuss with the class why one strategy can produce different scores.
Explore
15 Minute(s)
Display slide 9 and pass the attached Heads or Tails handout to each student pair. Then, give each pair a coin to flip or have students use the CPM Probability Generators (alternatively, let them imagine flipping a coin).
Tell students to write all possible outcomes when flipping a coin 3 times on their handout, as well as the probability of 0, 1, 2, or 3 heads occurring in the situation. To stretch their knowledge, have students create a histogram to display their data.
Circulate around the room to help students and answer questions. Be sure not to give students the possible outcomes, probabilities, or formal language yet. This is the time for students to demonstrate what they do and do not know. Make notes of questions and misunderstandings to address during the Explain phase.
After about 10 minutes, have some groups share the possible outcomes of flipping a coin three times. As students share their answers, fill the table on slide 9 with the answers for students to see as they complete their own tables.
Transition to slide 10 and complete the probability table. Then, ask students to reflect on the following questions:
Do you see a pattern? If so, describe it.
Are TTH and HTT the same outcome? How do you know if order matters in an outcome?
Have some groups share their answers.
Use the hidden slide 11 for sample responses to the Heads or Tails handout.
Once students complete their work, have them set aside their Heads or Tails handout. Students will check their answers during the Explain phase.
Explain
25 Minute(s)
Show slide 12 and pass out the attached Benefits handout to each student. Move through slides 12–14 and go over the definitions: random variables, discrete random variables, continuous random variables, and probability distribution. After each definition, ask students to come up with and share an example for that term.
Move to slide 15 and direct students’ attention to Example 1: Flipping a Fair Coin. Have students look back at their Heads or Tails handout and match their answers with the ones on the slide. Clarify any misconception if necessary and have them correct any wrong answer on their handout.
Emphasize to students that they already know what a random variable is, but they now have the formal academic language to use.
Transition through slides 16–19 to review with students how to create a probability distribution graph. This is the time to clear up any misconceptions and strengthen academic vocabulary.
Display slide 20 and ask students to predict what the notation is asking them to do. Once students share their answers, tell them that the notation is asking them to decide the probability of flipping a coin three times in which 1, 2, or 3 heads come up. Move through slides 21–23 to model how to calculate that probability.
On slide 24, review the possible values of P(X) and the sum of all probabilities in any given scenario.
Transition to slide 25 and direct students’ attention to the back of their handout. Introduce students to the formulas for mean (expected value) and standard deviation of a discrete random variable. Then, introduce students to the notation in each equation. Formally name the Greek letters—μ (mu) and σ (sigma)—so that students feel more comfortable reading and discussing the formulas.
Display slide 26 and read the following scenario to the class:
There is a deck of four cards: an ace of hearts, 2 of hearts, 3 of hearts, and an ace of spades. One card is randomly drawn, replaced, and a second card is drawn. Let X be the sum of the two drawn cards, where the ace has a value of 1.
Ask the class to create a probability distribution table and graph. Then give students a couple of minutes to discuss with their Elbow Partners what the first problem-solving steps are and then ask a few groups to share their plans with the class. Instruct student pairs to complete the sample space tables and probability distribution table.
Display slide 27 and have students check their work with what is on the slide. Encourage them to correct information as necessary.
Show slide 28 and ask pairs to now create a graph for their probability distribution table.
Once students complete their graphs, show slide 29 so that students can check their work.
Transition through slides 30–31 and demonstrate to students how to find the expected value and standard deviation using their probability distribution table.
Once students finish filling in their guided notes, have them keep their handouts in their notebooks for future reference.
Engage 2
10 Minute(s)
At the beginning of day 2, display slide 32 and introduce the class to the Monty Hall problem. Ask students to pick a door. Designate three walls in the classroom to represent each of the three doors (one wall per door) and ask students to stand against the wall that corresponds to the door they chose.
Once students have made their choice, display slide 33. Tell all students standing at door 2 that, if this were the real game “Let’s Make a Deal”, they would have lost. However, for this activity, they get to have an extra chance—have them choose a different door. Tell the other students that they can make a choice: either choose to switch doors or stay at the one they originally chose.
Send students back to their seats and transition through slides 34–37. Explain to students the statistics involved in answering this question and why it is statistically beneficial to switch doors instead of staying.
If time allows, play the “Monty Hall Problem” video by Numberphile on slide 38. If needed to save time, stop the video at 2:47 or 4:49.
Extend
10 Minute(s)
Display slide 39. Have students take out a piece of paper or give each student a copy of the attached Video Reflection Questions handout.
Tell students that they will watch a video to learn how underwriters use discrete random variables in their careers and answer the provided questions.
Show slide 40 and play the embedded ICAP video on the slide. Students will hear how a professional uses discrete random variables in their careers.
Show slide 41 and, as long as time allows, facilitate a whole class discussion on:
What discrete random variables does he mention?
How does he use statistics and/or probability in his job?
What two new things did you learn?
Ask students to share their answers.
Evaluate
30 Minute(s)
Display slide 42 and share the Choice Board instructional strategy with the class. Give each student a copy of the attached Choice Board handout. Instruct students to work independently to complete enough tasks to earn 6 points.
Once students have completed the Choice Board, use established classroom procedures to collect their work.
Resources
K20 Center. (n.d.). Choice Board. Strategies. https://learn.k20center.ou.edu/strategy/73
K20 Center. (n.d.). Elbow Partners. Strategies. https://learn.k20center.ou.edu/strategy/116
Numberphile. (2014, May 22). Monty hall problem [Video]. YouTube. https://youtu.be/4Lb-6rxZxx0
Numberphile. (2021, April 28). The math of being a greedy pig. [Video]. YouTube. https://youtu.be/ULhRLGzoXQ0
Minas, M. (2020, March 30). Greedy Pig. [Video]. YouTube. https://youtu.be/4KgRfUwpxPU
Skitterphoto. (2017, December 23). Brain, mind, psychology [Photograph]. Pexels. https://www.pexels.com/photo/pair-of-white-dice-on-top-of-mirror-705171/