[IS.3 - Struggling Learners] After this lesson, students will understand the difference between independent and dependent events and theoretical and experimental probability. They will be able to calculate compound probabilities as well as conditional probabilities. Students learn best when they can see a purpose for learning the content. Probability happens all around us, and if the examples are situations that students can relate to, their motivation to learn increases as well as their success in the unit concepts.
Introduce the game show “Let’s Make a Deal” to the class. [IS.4 - All Students]
“Suppose you are on a game show, and you are given the choice of three doors: behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what is behind the doors, opens another door, say No. 3, that has a goat. He then says to you, ‘Do you want to pick door No. 2?’Is it to your advantage to switch your choice?”
Go through a few demonstrations of the game and have students be the contestants. Use cardboard boxes with a prize in one of the boxes and two stuffed-animal goats (or pictures of a goat) in the other two. Make sure to mix up what is in each box after each contestant chooses. Tell students who are watching to jot down notes and brainstorm the probability of winning the prize.
After the demonstration, have a discussion about the probability of winning. Show the graphic that breaks down the individual outcomes with and without switching.
“After choosing a box, the host opens another one and gives you the choice to switch boxes. What is the probability that you will win the prize?” The common misconception behind this problem is that the probability is 1/2 and most likely students will say 1/2. The incorrect assumption is that there are two remaining doors when there are actually three and the probability is 1/3.
Show the Monty Hall Problem video on YouTube: https://www.youtube.com/watch?v=mhlc7peGlGg.
Discuss their reactions to the problem and the video. They can also see a demonstration at http://demonstrations.wolfram.com/MontyHallProblem/.
Students should put the following vocabulary into their notes. [IS.5 - Struggling Learners] The vocabulary is from http://mathworld.wolfram.com.
An event is the outcome of an experiment. For example, flipping a coin results in two events: heads or tails. [IS.6 - All Students]
P(A) is the notation for the probability of event A occurring.
1 − P(A) is the notation for the probability of event A not occurring.
P(A ∩ B) is the notation for the probability of event A and event B occurring.
P(A ∩ B) = P(A) × P(B)
P(A U B) is the notation for the probability of event A or event B occurring.
P(A U B) = P(A) + P(B) − P(A ∩ B)
Two events are independent if one event occurs and it does not affect the probability of the other event. [IS.7 - Struggling Learners] For example, if you flip a coin twice, getting a heads on the first flip does not affect the probability of getting a tails on the second flip. It’s still 1/2.
This means that if two events are dependent, then the occurrence of one event affects the probability of the other event. For example, the probability of you picking a red card from a deck of 52 cards is 26/52 = 0.5 (since half the cards in the deck are red). If you keep that card out, the probability of you picking another red card is now 25/51 = 0.49 because the card you’ve taken out is one of the red cards (26 – 1) and one of the total 52 cards (52 – 1).
The Monty Hall problem is an example of conditional probability. Conditional probability is the probability of one event happening given another event has already occurred.
P(A|B) is the notation for the probability of event A occurring given event B has already occurred: P(A|B) = P(A ∩ B) ÷ P(B).
Theoretical probability is the predicted outcome, whereas the experimental probability is the actual outcome. For example, the theoretical probability of rolling a 4 on a number cube is 1/6 = 0.167 (since the 4 appears on one of the six sides). If you performed an experiment and you rolled a 4 once out of ten rolls or 1/10 = 0.10, this is the experimental probability.
The Fundamental Counting Principle is a method used to calculate all the possible combinations of a given event. For example, if you are getting ready for school and you have three different pants to wear and five different shirts to wear, you have 3 × 5 or 15 different outfits.
The product of an integer and all the integers less than it and greater than zero is called a factorial. It is denoted “n!.” For example, 4! = 4 × 3 × 2 × 1 = 24. Notice that there is one way to arrange zero objects and that expression as a factorial is 0! = 1.
A permutation is the number of ways things can be arranged in order. For example, if Alex, Ben, and Colleen are running for student council president, vice-president, and treasurer, there are 6 ways to arrange them in the student council: ABC, ACB, BAC, BCA, CAB, and CBA. Think of the p in permutation as representing position.
A combination is the number of ways things can be arranged where order does not matter. For example, if Alex, Ben, and Colleen are trying out for 2 spots on a sports team, there are only 3 ways they can make the team: AB, AC or BC. Alex and Ben making the team is the same as Ben and Alex making the team (order does not matter). Think of the c in combination as representing committee.
Activity 1: Pairs
If possible, set students up in a computer lab or have students use laptops. If this is not possible, then you can do this activity as a whole class. [IS.8 - Struggling Learners] Hand out the Spinner Activity Worksheet (M-A2-1-1_Spinner Activity Worksheet.doc).
Go to http://illuminations.nctm.org/ActivityDetail.aspx?ID=79. It has a spinner on it that can be adjusted for the number of sectors. Instead of using the Web site’s numbers, have students compute the theoretical and experimental probability.
Activity 2: The Birthday Problem
Ask students to think about the probability of any two students in class having their birthdays on the same day of the week, (i.e., Monday, Tuesday, etc.).
“The probability that two students have a birthday on the same day of the week is (7/7) × (1/7) = .14285 . You start with 7/7 because the first student can have his/her birthday on any day of the week, and you multiply by 1/7 because the second student must have his/her birthday on the same day of the week. To get the total probability, the probabilities of both events are multiplied together. Sometimes working with the probability of an event not happening is easier to calculate than the probability of an event happening. For example, the probability that two people do not have the same birthday is (365/365) × (364/365) = 0.99726. The probability of the first student’s birthday is 365/365, as the student’s birthday can be any day of the year. The probability of the second student’s birthday is 364/365, as it can be any day except one. Therefore, the probability of two people having the same birthday is 1 − 0.99726 = 0.00274.”
Go to http://demonstrations.wolfram.com/TheBirthdayProblem/ and show students the demonstration. [IS.9 - All Students] Change the number of people and also use the number of students in the classroom. This gives them the theoretical probability. Then have students calculate the experimental probability.
Activity 3: Think-Pair-Share
Discuss sample size with students. In the following problem, the sample size starts with 20 candies. Each time a candy is taken, the sample size is reduced by one unless it is replaced. The concept of replacement is important and should be emphasized with reference to an experiment. A bag contains one blue and one red marble. Randomly drawing the blue marble has a probability of 0.5. Not replacing the blue marble for the next random draw means that the probability of drawing the red marble is 1, and the probability of drawing the blue marble is 0. However, if you replace the blue marble after the first draw, the probability is the same as for the original draw, 0.5. Point out that these types of activities are different from coin flips or rolls of one or more number cubes because replacement does not apply to them. Also point out the difference between a flip and a roll. One coin flip has 2 outcomes; one roll has 6 outcomes.
Present to students the following problem: “You are sitting in front of the principal’s desk when you notice she has a jar of candy. [IS.10 - All Students] You count 10 strawberry-flavored candies, 5 vanilla-flavored, 4 blueberry-flavored, and 1 chocolate-flavored in the jar.”
1. What is the probability of each flavor? [IS.11 - Struggling Learners]
P(S) = 10/20 P(V) = 5/20
P(B) = 4/20 P(C) = 1/20
2. The principal tells you that you may randomly pick only one candy. What is the probability of picking a blueberry or a strawberry? P(B U S) =
P(B) + P(S) = ( 4/20 + 10/20 ) = 14/20
3a. She tells you that you may randomly pick two candies. After picking the first candy, it is placed back in the jar. What is the probability of picking one vanilla and one chocolate?
P(V ∩ C) =( 5/20 × 1/20 ) = 5/400 = 1/80 = 0.0125
3b. What is the probability of picking one vanilla and one chocolate if the first candy is not placed back in the jar? (Remember that the sample size, the denominator, is reduced by one for the second candy. The first product is the probability of picking a vanilla first and then a chocolate. The second product is the probability of pulling a chocolate first and then a vanilla.)
( 5/20 × 1/19 ) + ( 1/20 × 5/19 ) = 5/380 + 5/380 = 10/380 = 0.02632
4a. What is the probability of randomly picking one blueberry given that you have already picked a vanilla candy and replaced it in the jar?
P(B|V) = ( 4 blueberry/20 candies ) = (0.2)
4b. What is the probability of randomly picking one blueberry given that you have already picked a vanilla candy and have not replaced it?
P(B|V) = ( 4 blueberry/19 candies ) = (4/19).
5. Let’s say that each candy had a number on it in the jar. The ten strawberry candies were numbered 1 through 10, the five vanilla candies were numbered 1 through 5, the four blueberry candies were numbered 1 through 4 and the chocolate was numbered 1. Using the fundamental counting principle, how many ways can you pick one of each color? (200)
Activity 4: Groups
Divide the class into groups of four. [IS.12 - All Students] Give them each a deck of cards. Allow them 10 minutes to write ten probability questions regarding drawing from the deck. For example, what is the probability of picking one red card? What is the probability of picking a red card and a four? After they have written their questions, have each group share one with the whole class and write them on the board. When ten questions are on the board, the groups have to solve the ten questions. Review and discuss the answers as a whole class. Collect each group’s questions and use them as a warm-up the next day or on a quiz.
An Exit Ticket (M-A2-1-1_Lesson 1 Exit Ticket.doc and M-A2-1-1_Lesson 1 Exit Ticket KEY.doc) is a quick way to evaluate whether students understand the concepts.
Use the following strategy to tailor the lesson to meet the needs of your students throughout the year.
This lesson is a building block for the lessons to come and is necessary for students to understand probability and how it is used in real-world situations. It begins with a game show activity that engages students and piques their interests. The teacher then introduces some vocabulary that students will need to know before the activities. The activities are simple and provide students time to explore probability. Students are given time to review the lesson’s concepts and are given timely feedback. The teacher also explains how students will use this information in the next lesson.