Probability and Predictions
Probability and Predictions
Objectives
Students will become acquainted with probability informally through experiments. Students will be able to understand and apply basic concepts of probability. They will understand and use appropriate terminology to describe elementary probabilities and experimental results. Using proportionality and a basic understanding of probability, students will test conjectures about the results of experiments and simulations. Students will:
- determine and understand experimental and theoretical probabilities.
- reason using probabilities associated with experiments.
- develop an understanding of the vocabulary necessary to communicate the concepts of probability.
- develop ways to identify the possible outcomes of an event.
- investigate elementary probabilities experimentally and theoretically, and understand the differences in calculation.
- make predictions using experimental and theoretical probability.
- 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
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?
How does the type of data influence the choice of display?
In what ways are the mathematical attributes of objects or processes measured, calculated and/or interpreted?
What does it mean to estimate or analyze numerical quantities?
What makes a tool and/or strategy appropriate for a given task?
- What does it mean to estimate or analyze numerical quantities?
- What makes a tool and/or strategy appropriate for a given task?
- How can data be organized and represented to provide insight into the relationship between quantities?
- How does the type of data influence the choice of display?
- How can probability and data analysis be used to make predictions?
- In what ways are the mathematical attributes of objects or processes measured, calculated and/or interpreted?
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.
- http://www.shodor.org/educators/activities/
- http://www.shodor.org/interactivate/tools/
- http://www.random.org
(random number, coin, card, generators) - http://roll-dice-online.com
(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
(generates 1,000 rolls for the sum of two (or three) number cubes with each push of a button, for thousands of rolls in seconds)
- http://www.saintannsny.org/depart/math/misterg.htm
(rolls two number cubes and creates a bar graph of the results) - http://nces.ed.gov/nceskids/chances/dice_handler.asp
(NCTS; also see the activities sections of this Web site) - http://teacher.scholastic.com/lessonrepro/lessonplans/grmagam.htm (description of some simple probability games: two-number-cube sums game, game of pig)
- http://www-stat.stanford.edu/~susan/surprise/
(various applets including the birthday generator and “traffic jam”) - http://www.shodor.org/interactivate/activities/Marbles/
- http://archon.educ.kent.edu/Oasis/Resc/Educ/data.html
- NCTM Standards and Expectations for Grades 6-8, Data Analysis and Probability
Formative Assessment
-
View
Multiple-Choice Items:
For questions 1 and 2, use the experimental data represented in the frequency table below.
- Find P(even) =
A 0.4
B
C
D
- Find P(not 4) =
A
B 10%
C
D
- Which of the following is not equally likely?
A passing or failing a quiz
B flipping a penny and getting heads or getting tails
C rolling an odd or an even on a six-sided number cube
D drawing a heart or a diamond in a standard deck of cards
- A bag contains 6 red, 8 blue, and 11 yellow marbles. What is the probability of selecting blue when one marble is drawn randomly?
A 32%
B
C 0.375
D 0.48
For questions 5 and 6, use the spinner at the right.
- What is the theoretical probability of getting pink
in one spin of the spinner?
A 0.24
B 25%
C 50%
D
- What is the theoretical probability of getting green twice in two spins of the spinner?
A
B 0.375
C 50%
D
For questions 7 and 8, use the tree diagram below.
- At a school carnival, a spinner game awards a large stuffed animal to every person who gets green on spinner 1 and an odd number on spinner 2. There are 280 people expected to play the game. How many large stuffed animals will be needed?
A 20
B 28
C 34
D 35
- The carnival game represented by the tree diagram gives prizes to every player who does not spin a red or an even number. There are 344 people expected to play the game. How many people are expected to win some sort of prize?
A 102
B 129
C 138
D 144
Use the table below to answer question 9.
- The table indicates the number of customers at the Corner Pie Shoppe who ordered specific combinations of its most popular pies with ice cream. The store usually sells an average of 435 slices each weekend. How many slices of peach pie should be prepared?
A 145
B 174
C 192
D 240
Multiple-Choice Answer Key:
1. D
2. D
3. A
4. A
5. C
6. A
7. D
8. B
9. B
Short-Answer Items:
- To help determine her allowance each week, Jaelynn’s parents put several blocks marked with different dollar values in a bag. The values were $3.50, $5.00, $8.00, and $10.00. Each week she is asked to randomly draw a block from the bag and then return it. After the first 20 weeks, Jaelynn realized that she had drawn $3.50 two times, $5.00 eight times, $8.00 six times, and $10.00 four times. Based on what Jaelynn has drawn so far, find the probability of drawing each amount.
- Use Jaelynn’s experimental data from problem 10. Jaelynn’s parents revealed to her that there were 36 blocks in the bag. Predict how many blocks of each dollar value are actually in the bag.
- Consider a game in which you roll a six-sided number cube and flip a coin.
Find the theoretical probability of not rolling an even number and getting heads. Show all your work.
Short-Answer Key and Scoring Rubrics:
10. To help determine her allowance each week, Jaelynn’s parents put several blocks marked with different dollar values in a bag. The values were $3.50, $5.00, $8.00, and $10.00. Each week she is asked to randomly draw a block from the bag and then return it. After the first 20 weeks, Jaelynn realized that she had drawn $3.50 two times, $5.00 eight times, $8.00 six times, and $10.00 four times. Based on what Jaelynn has drawn so far, find the probability of drawing each amount.
P($3.50)=
or 10%P($5.00)=
or 40%P($8.00)=
or 30%P($10.00)=
or 20%(check: 10% + 40% + 30% + 20% = 100%)
Points
Description
2
- Written explanation or work is complete and detailed.
- The response demonstrates thorough understanding of experimental probability.
- Answers are all mathematically correct and in proper form.
1
- Written explanation or work is correct but brief or simplistic.
- The response demonstrates partial understanding of experimental probability.
- Most answers are mathematically correct but may not be in simplest form, or labels are missing.
0
- Written explanation or work is incorrect or missing.
- The response demonstrates no understanding of experimental probability.
- Most answers are incorrect or missing.
11. Use Jaelynn’s experimental data from problem 10. Jaelynn’s parents revealed to her that there were 36 blocks in the bag. Predict how many blocks of each dollar value are actually in the bag.
Since the probability of each value can be written as a fraction or decimal, 36 blocks can be multiplied by each probability to predict the number of each type of block.
$3.50 was 10% = 0.1 so 36 × 0.1 = 3.6 or about 4 blocks
$5.00 was 40% = 0.4 so 36 × 0.4 = 14.4 or about 14 blocks
$8.00 was 30% = 0.3 so 36 × 0.3 = 10.8 or about 11 blocks
$10.00 was 20% = 0.2 so 36 × 0.2 = 7.2 or about 7 blocks
(check 4 + 14 + 11 + 7 = 36 total)
(Students could also choose to set up proportions to solve the problem.)
Points
Description
2
- Written explanation or work is complete and detailed.
- The response demonstrates thorough understanding of making predictions based on probability.
- Answers are all mathematically correct and in proper form.
1
- Written explanation or work is correct but brief or simplistic.
- The response demonstrates partial understanding of making predictions based on probability.
- Most answers are mathematically correct but may not be in simplest form, or labels are missing.
0
- Written explanation or work is incorrect or missing.
- The response demonstrates no understanding of making predictions based on probability.
- Most answers are incorrect or missing.
- Consider a game in which you roll a six-sided number cube and flip a coin.
Find the theoretical probability of not rolling an even number and getting heads. Show all your work.
Sample space may vary. Tree diagram may be used.
1
2
3
4
5
6
H
1H
2H
3H
4H
5H
6H
T
1T
2T
3T
4T
5T
6T
P(not even, heads) =
or 25%Points
Description
2
- Written explanation includes a detailed sample space.
- The response demonstrates thorough understanding of theoretical probability.
- Answer is mathematically correct and in proper form.
1
- Written explanation or work is correct, but a sample space is not used.
- The response demonstrates partial understanding of theoretical probability.
- Answer is mathematically correct or very close, but may not be in simplest form or is missing a label such as the % sign.
0
- Written explanation or sample space is incorrect or missing.
- The response demonstrates no understanding of theoretical probability.
- Answers are incorrect or missing.
Performance Assessment:
Spin Off
Ben and his sister plan to play a spin-off game to determine who must wash the dishes each evening for a month. Complete the steps below.
- On a separate piece of paper, design a spinner with three colors. The probability of landing on each color with one spin should be:
Red–50%
Yellow–25%
Black–25%
- On a separate piece of paper, use a table, tree, or organized list to display the sample space of outcomes for spinning this spinner twice.
- Use the sample space (from question 2) to find the probabilities below:
a) P(red/red or black/black) =
b) P(nonmatching colors without yellow) =
c) P(yellow/yellow) =
- Scoring your Spin Off:
- Each turn consists of two spins.
- A player earns 1 point on his/her turn if s/he spins twice and gets either two red or two black.
- A player earns 1 point on his/her turn if s/he spins twice and gets one each of black and red.
- If either player spins two yellows, s/he loses 2 points.
- If a player gets anything else, s/he is done with his/her turn and does not gain or lose points.
Ben and his sister will each have 20 turns. Considering the scoring system, predict how many times Ben and his sister will each lose and have to wash the dishes. Show or explain why.
- Use the spinner you created to take twenty turns for each player and record the results in the scoring table below.
Turn #
Player 1
Player 2
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Total Score
- Compare your predicted results (question 4) and your experimental results (question 5). Discuss whether Ben and his sister are equally likely to win, and how many trials would be needed to be confident in your prediction.
Performance Assessment Scoring Rubric:
Points
Description
4
- Spinner is original with each color accurately represented. Spinner has a total of four or more sections.
- Response uses one or more appropriate sample space organizer(s) to accurately list all possible outcomes.
- Response accurately calculates all three theoretical probabilities, with each written in more than one form (fraction, decimal, or percentage).
- Response demonstrates excellent understanding of the mathematical ideas and processes related to probability and prediction.
- Written comparison is thorough, detailed, and insightful.
- Response works beyond the problem requirements, possibly by checking steps and/or incorporating technology.
3
- Spinner accurately represents each color. Spinner has a total of three or four sections.
- Response uses an appropriate sample space organizer to accurately list all possible outcomes.
- Response accurately calculates all three theoretical probabilities, with each represented in one form (fraction, decimal, or percentage).
- Response demonstrates substantial understanding of the mathematical ideas and processes related to probability and prediction.
- Written comparison is thorough.
- Response meets all of the problem requirements.
2
- Spinner includes each of the three colors, but the size of the sections is not accurate.
- Sample space organizer has minor errors which create inaccuracy in the list of possible outcomes.
- Response accurately calculates two of three theoretical probabilities.
- Response demonstrates some understanding of the mathematical ideas and processes related to probability and prediction.
- Written comparison is brief.
- Response partially meets the problem requirements.
1
- Spinner is present but inaccurate.
- Sample space is disorganized and incomplete.
- Response contains major errors in the calculation of theoretical probabilities.
- Response demonstrates substantial lack of understanding of portions of the problem.
- Written comparison is incomplete or has inaccuracies throughout.
- Response does not meet several of the problem requirements.
0
- Spinner is unfinished or missing.
- No sample space organizer is used, and the list of outcomes is missing or has serious errors.
- Theoretical probabilities are all incorrect or missing.
- Response demonstrates a complete lack of understanding of the problem.
- Written comparison is missing.
- Response does not meet any of the problem requirements.
Final 06/07/2013
