Probabilities of Simple and Compound Experiments
Probabilities of Simple and Compound Experiments
Objectives
Students will understand and determine theoretical probabilities. They will reason using theoretical probabilities associated with experiments. Students will:
- develop an understanding of the vocabulary terms necessary to communicate the concepts of probability.
- develop ways to identify the possible outcomes of an event.
- develop strategies for analyzing probabilities using lists, trees, counting, and area models.
- understand that when all outcomes of an experiment are equally likely, the theoretical probability of an event is the fraction of the outcomes in which the event occurs.
- analyze situations that involve independent events.
- use experimental and theoretical probability and proportions to make predictions.
- discover that the greater the number of trials in a random experiment, the closer the experimental probability will be to the theoretical probability (the law of large numbers).
Essential Questions
- What makes a tool and/or strategy appropriate for a given task?
- In what ways are the mathematical attributes of objects or processes measured, calculated, and/or interpreted?
- How can data be organized and represented to provide insight into the relationship between quantities?
- How can probability and data analysis be used to make predictions?
Related Unit and Lesson Plans
Related Materials & Resources
The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.
- Find Random Reporter here: http://www.portal.state.pa.us/portal/server.pt?open=space&name=Dir&psname=SearchResult&psid=2&cached=true&in_hi_userid=2&control=OpenSubFolder&subfolderID=3267&DirMode=1
- http://www.shodor.org/educators/activities/
- http://www.shodor.org/interactivate/tools/
- Random number, coin, card generators
- Sophisticated and flexible random-number generator and number-cube roller
- http://www.betweenwaters.com/probab/probab.html
- http://www.betweenwaters.com/probab/dice/dicemain3D.html
- Web site that creates all kinds of graphs
http://nces.ed.gov/nceskids/createagraph/default.aspx
- Numerous links to math puzzles and games
- Web site that rolls two number cubes and creates a bar graph of the results
http://www.sai ntannsny.org/depart/math/misterg.htm
- http://www.shodor.org/interactivate/lessons/HistogramsBarGraph/
- NCTS; also see the activities sections of this Web site
http://nces.ed.gov/nceskids/chances/dice_handler.asp
- Description of some simple probability games: two number-cube sums game, game of pig
http://teacher.scholastic.com/lessonrepro/lessonplans/grmagam.htm
- Various applets including the birthday generator and “traffic jam”
Formative Assessment
-
View
Multiple-Choice Items:
- Use the table below to answer the question.
Maurice’s Candy Bag
Flavor of Candy
Number Pulled
Butterscotch
4
Cherry
2
Lemon
9
Peppermint
5
Maurice has a bag with 100 wrapped candies that are all the same shape and size. He pulled out one candy at random, returned it to the bag, and then did it again. He repeated this 20 times and recorded the results in the table. Based on Maurice’s experimental data, which is most likely the number of cherry candies in the bag?
A
2
B
10
C
20
D
25
- Which of the following two outcomes are equally likely on one roll of a six-sided number cube?
A
multiple of 3
multiple of 2
B
number less than 2
number greater than 4
C
number 4
odd number
D
multiple of 2
odd number
Use the picture of the spinner below to answer question 3.
- Jerry spun the spinner 15 times and made a line plot with the data. Which line plot most likely represents Jerry’s spins?
Use the table below to answer question 4.
Name
Points Earned
Ashlynn
520
Monica
205
Perry
190
Tony
85
- Four friends played a dart game. They took turns throwing darts at the board. Each dart hit the board randomly. The dart board had each person’s initial on it. When players hit the section with their initial, they scored 5 points. Which is most likely the board that the four friends used?
Use the tree diagram below to answer questions 5 and 6.
Sylvie’s favorite restaurant, Bandito, offers burritos with two choices of tortilla and three choices of filling. Sylvie made a tree diagram of all the possible types of burritos she could order.
- When Sylvie ordered a burrito one day, she told the server to surprise her and choose the tortilla and filling. What is the probability that Sylvie got a burrito with a flour tortilla and either pork or steak?
A
B
C
D
- Bandito added a new rice filling to its choices. How many different burritos can be made at Bandito with four filling choices?
A
4
B
8
C
10
D
16
- Darlene and Stephen played a game with two number cubes. They rolled the two cubes and multiplied the numbers together. Which rule would be fair for both Darlene and Stephen?
×
1
2
3
4
5
6
1
1
2
3
4
5
6
2
2
4
6
8
10
12
3
3
6
9
12
15
18
4
4
8
12
16
20
24
5
5
10
15
20
25
30
6
6
12
18
24
30
36
A
Darlene gets a point if the product is even and Stephen gets a point if the product is odd.
B
Darlene gets a point if the product is a multiple of 2 and Stephen gets a point if the product is a multiple of 3.
C
Darlene gets a point if the product is 12 and Stephen gets a point if the product is 16.
D
Darlene gets a point if the product is one or a prime number, and Stephen gets a point if the product is 24 or greater.
- Tanya is a basketball player. Tanya made an area model based on her free-throw average. Which is most likely Tanya’s area model for two free-throws if her free-throw average is
?
- What is the definition of an independent event?
A
an event such that the outcome of the first event has no effect on the probability of the second event
B
an event that consists of two or more simple events
C
the opposite of an event; the set of all outcomes of an experiment that are not included in the event
D
modeling a real event without actually observing the event
Multiple-Choice Answer Key:
1. B
2. D
3. C
4. D
5. B
6. B
7. D
8. C
9. A
Short-Answer Items:
- How does the law of large numbers relate to experimental probability? Give an example that supports your answer.
- Tricia played a game with two spinners. One was a number spinner; the other was a color spinner. She made the organizer chart below to show the sample space for spinning each spinner once.
Design the number spinner and the color spinner Tricia used, and explain your design.
- Jon’s mother gives him $7 each week for washing the dishes. One week she would let Jon choose one bill from jar 1 and one bill from jar 2 without looking. Jar 1 contains one $20 bill and four $1 bills. Jar 2 contains one $5 bill and five $1 bills.
- List the outcomes of the sample space.
- List the probabilities of each outcome.
- What is the probability of Jon making more than $7?
- Would Jon be better off taking the original $7? Explain your answer.
Short-Answer Key and Scoring Rubrics:
- How does the law of large numbers relate to experimental probability? Give an example that supports your answer.
Answers will vary. Students should state that the larger the number of experiments, the closer the experimental probabilities for the outcomes will be to the theoretical probability.
Points
Description
2
- The written explanation is complete and detailed.
- The explanation demonstrates thorough understanding of the law of large numbers.
- The explanation is supported with an example or visual.
1
- The written explanation is correct but brief or simplistic.
- The explanation demonstrates partial understanding of the law of large numbers.
- The student attempts to support the explanation with an example.
0
- The written explanation is incorrect or missing.
- The explanation demonstrates no understanding of the law of large numbers.
- The student does not support the explanation at all.
- Tricia played a game with two spinners. One was a number spinner; the other was a color spinner. She made the organizer chart below to show the sample space for spinning each spinner once.
Design the number spinner and the color spinner Tricia used, and explain your design.
Explanations will vary, but the spinners should look as below.
Points
Description
2
- Two spinners correctly reflect the data.
- Both spinners are completely and accurately labeled.
- There is a detailed explanation to support the visual representation.
- The student shows complete understanding of the mathematics.
1
- Two spinners correctly reflect the data, or with slight errors in spacing.
- Both spinners are labeled with slight errors, or only partially labeled.
- Little work or explanation supports the visual representation.
- The student shows partial understanding of the mathematics.
0
- Both spinners are incorrectly drawn or are missing.
- Both spinners have incorrect labels or labels missing.
- No work or explanation is provided to support the visual representation.
- The student shows no understanding of the mathematics.
- Jon’s mother gives him $7 each week for washing the dishes. One week she would let Jon choose one bill from jar 1 and one bill from jar 2 without looking. Jar 1 contains one $20 bill and four $1 bills. Jar 2 contains one $5 bill and five $1 bills.
- List the outcomes of the sample space.
- List the probabilities of each outcome.
- What is the probability of Jon making more than $7?
- Would Jon be better off taking the original $7? Explain your answer.
Jon would be better off taking his original allowance. His probability of receiving more than $7 by choosing from the jars is 20% and receiving less is 80%. Supporting answers will vary. It is interesting to note that Jon’s long-range expected value from taking the new offer is $6.47, very close to (but still less than) the original allowance. (This is a good way to launch expected value.)
Points
Description
2
- The written explanation is thorough, clear, and supported with visual representation.
- The answer shows complete understanding of sample space and probability.
- The answer is correct.
- The answer meets the requirements of the problem.
1
- The written explanation is brief and is partly supported by the visual representation.
- The answer shows partial understanding of sample space and probability.
- The answer is incorrect due to a minor mathematical error.
- The answer partially meets the requirements of the problem.
0
- The written explanation is brief or missing and there is no visual representation.
- The answer shows no understanding of sample space and probability.
- The answer is incorrect due to a major mathematical error / misunderstanding or is missing.
- The answer does not meet the requirements of the problem.
Performance Assessment:
The Performance Assessment should be completed in groups of two to three students.
You are in charge of setting up the games at a carnival and you have to create games that will entice people to play but still make money (in other words, they appear fair but are actually in the carnival’s favor to win). You will design an original game that incorporates a variety of different outcomes. Write the rules and points scored for each specific outcome. Complete all of the following steps:
- Your game should include more than one stage (for instance, spinning a spinner and rolling a number cube or drawing objects from three bags, etc.). You must have multiple stages in your game to get the full score (4) on your assessment.
- Your game must be fair or very close to fair, so that somebody playing it has a chance to win. If your game is fair, prove how. If your game is not fair, show how it is very close. If your game would involve strategy to win, explain the best strategy and how you determined this. You must include a detailed description of the game and the rules for playing it.
- List all of the possible outcomes for playing your game in the most appropriate kind of organizer (list, tree, sample-space chart).
- Give all of the probabilities for the various outcomes and any specific events if you have conditions for winning.
- Play the game a number of times and collect the experimental probabilities for the outcomes. Write a detailed comparison of the theoretical and experimental probabilities.
- Pick a partner group. Play your partner group’s game and have them play your game. Write a comparison of the two games.
- Write one or more paragraphs describing what you have learned in this unit and what designing the game for this project taught you.
- Write one or more paragraphs describing any questions you still have or extensions of the lessons you would like to learn more about.
Performance Assessment Scoring Rubric:
Points
Description
4
- The game is sophisticated and original with a detailed description of the game and rules. The game has compound events or involves strategy and may have an extension version.
- The students use appropriate methods for listing outcomes, and probabilities have no mathematical errors.
- The students demonstrate advanced understanding of the questions, mathematical ideas, and processes.
- The students work beyond the problem requirements. The game shows creativity and possibly incorporates technology.
- The students’ data gathered by playing their own game and another group’s game is extensive and organized.
- The students’ written report is thorough, detailed, and professional. Self-analysis is insightful and includes two or more questions or extension ideas.
3
- The game is original with an adequate description of the game and rules. Strategy or compound outcomes are simple or too complicated.
- The students list outcomes correctly, and probabilities have no major mathematical errors.
- The students demonstrate substantial understanding of the questions, mathematical ideas, and processes.
- The students meet all of the problem requirements.
- The students’ data gathered by playing their own game and another group’s game is organized and complete. The comparison is adequate.
- The students’ written report is thorough. Self-analysis is complete and includes at least one question or extension idea.
2
- The game mirrors activities from class (not unique). It includes a brief description of the game and rules.
- There are several minor mathematical errors, or one serious flaw in reasoning when finding the probabilities.
- The students demonstrate some understanding of the questions, mathematical ideas, and processes.
- The students partially meet the problem requirements.
- The students’ data gathered by playing their own game and another group’s game is insufficient to draw conclusions. The comparison is brief.
- The students’ written report is brief. Self-analysis does not address both elements. No questions are asked or an unrelated question is asked.
1
- The game and/or rules are unoriginal or unfinished, or the rules do not make sense.
- The outcomes and/or probabilities are incorrect or missing.
- The students demonstrate substantial lack of understanding of portions of the problem.
- The students do not meet several of the problem requirements.
- The students’ data gathered by playing their own game and another group’s game is poorly organized or partially incomplete. The comparison is brief or missing.
- The students’ written report is unfinished. Self-analysis is missing or incomplete.
0
- The game and/or rules are unfinished or missing.
- Outcomes and/or probabilities are missing or have serious errors.
- The students demonstrate a complete lack of understanding of the problem.
- The students do not meet any of the problem requirements.
- The students’ data gathered by playing their own game is poorly organized or insufficient to draw conclusions. Data for the other group’s game is insufficient or missing. The comparison is missing.
- The students’ written report, including self-analysis, is missing.
Final 07/19/2013
