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Investigate whether a coin is fair by tossing it 1,000 times and comparing the percentage of heads to the theoretical probability 0.5.Clarifications
Clarification 1: Instruction includes representing probability as a fraction, percentage or decimal.Clarification 2: Instruction includes recognizing that experimental probabilities may differ from theoretical probabilities due to random variation. As the number of repetitions increases experimental probabilities will typically better approximate the theoretical probabilities.
Clarification 3: Experiments include tossing a fair coin, rolling a fair die, picking a card randomly from a deck, picking marbles randomly from a bag and spinning a fair spinner.
Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Event
- Experimental Probability
- Simulation
- Theoretical Probability
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
In grade 7, students use a simulation of a simple experiment to find experimental probabilities and compare them to theoretical probabilities. In grade 8, students will solve real-world problems involving probabilities related to single or repeated experiments, including making predictions based on theoretical probability.- Instruction includes opportunities for students to run various numbers of trials to discover that the increased repetition of the experiment will bring the experimental probability closer to the theoretical. Use virtual simulations to quickly show higher and higher volumes of repetition that would be difficult to create with physical manipulatives (MTR.5.1).
- Remind students that chance has no memory and each repetition in the simulation has the same probability distribution for the possible events.
- For example, if you flip a coin and it lands on heads, the next flip does not rely on the first outcome and still can be either heads or tails.
- Instruction focuses on the simple experiments listed in Clarification 3.
- For example, students can roll a 6-sided die 30 times to determine the experimental probability of “not rolling a 2.” Students can then compare their experimental probability to the theoretical probability of “not rolling a 2,” which is .
Common Misconceptions or Errors
- Students may incorrectly assume the theoretical and experimental probabilities of the same experiment will always be the same. To address this misconception, provide multiple opportunities for students to experience simulations of different situations, with physical or virtual manipulatives, in order to find and compare the experimental and theoretical probabilities.
- Students may incorrectly expect to see every possible outcome occur during a simulation. While all may occur in a simulation, it is not certain to happen. Students may inadvertently let their own experience with an experiment affect their response.
- For example, during an experiment if a student never draws an ace from a standard deck of cards, this does not indicate it could never happen.
Strategies to Support Tiered Instruction
- Teacher reviews the root words theoretical (theory) and experimental (experiment) and discusses the difference between a theoretical probability and experimental probability. Teacher provides graphic organizer to keep as reference for root words.
- For example, experimental probabilities are from simulations whereas theoretical probabilities are from calculations.
- Teacher provides opportunities to see a variety of outcomes.
- For example, open a deck of cards and draw 5 random cards. After looking at the 5 cards, discuss all the possible cards that could have been drawn but were not. This will help students see that not all possible outcomes will occur when an experiment is done.
- Teacher provides multiple examples for students to discuss if a probability in the example is theoretical or experimental. After each answer, students discuss how they know it is theoretical or experimental.
- For example, if one tosses a fair coin, the theoretical probability of landing on heads is 0.5. If one tosses a fair coin 14 times and it lands on hands 9 times, the experimental probability of landing on heads is based on the simulation.
- Teacher provides multiple opportunities for students to experience simulations of different situations, with physical or virtual manipulatives, in order to find and compare the experimental and theoretical probabilities.
Instructional Tasks
Instructional Task 1 (MTR.4.1)Each set of partners has been given a bag containing 5 red, 5 green, 5 yellow and 10 brown candies.
- Part A. Determine the theoretical probability for selecting one red candy at random from the bag. Do the same for blue, yellow and brown.
- Part B. Experimental Trials: Select one candy from the bag, record its color in the table below and return it to the bag. Repeat this process for a total of 20 trials.
- Part C. In the original table, record the total frequency of each color based on your 20 trials. Then calculate the experimental probability for each. How do the theoretical and experimental probabilities compare?
- Part D. Collect the data from 2 other sets of partners and combine your total frequencies. Complete the table below based on those 60 trials. How do the theoretical and experimental probabilities compare? How does that compare to your original calculations using 20 trials?
- Part E. Collect all of the class data to calculate new total frequencies and complete the table below.
- Part F.
- How do the theoretical and experimental probabilities compare?
- How does that compare to the calculations using 20 trials and 60 trials?
- What conclusions can you make about theoretical and experimental probabilities based on this information?
Instructional Items
Instructional Item 1A bag contains green marbles and purple marbles. If a marble is randomly selected from the bag, the probability that it is green is 0.6 and the probability that it is purple is 0.4. Dylan draws a marble from the bag, notes its color, and returns it to the bag. He does this 50 times. Approximately, how many times would you expect Dylan to draw a green marble?
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.
Related Courses
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Formative Assessments
Lesson Plans
Perspectives Video: Experts
Problem-Solving Tasks
Text Resource
Tutorials
MFAS Formative Assessments
Students are asked to determine the probability of a chance event and explain possible causes for the difference between the probability and observed frequencies.
Students are asked to estimate the probability of a chance event based on observed frequencies.
Students are asked to determine probabilities based on observed outcomes and to determine if the outcomes appear to be equally likely.
Students are asked to determine probabilities based on observed outcomes and to determine if the outcomes appear to be equally likely.
Student Resources
Problem-Solving Tasks
The purpose of this task is to provide students with the opportunity to determine experimental probabilities by collecting data. The cylindrical objects used in this task typically have three different resting positions but not all of these may be equally likely and some may be extremely unlikely or impossible when the object is tossed. Furthermore, obtaining the probabilities of the outcomes is perhaps only possible through the use of long-run relative frequencies. This is because these cylinders do not have the same types of symmetries as objects that are often used as dice, such as cubes or tetrahedrons, where each outcome is equally likely.
Type: Problem-Solving Task
This task introduces the fundamental statistical ideas of using data summaries (statistics) from random samples to draw inferences (reasoned conclusions) about population characteristics (parameters). In the task built around an election poll scenario, the population is the entire seventh grade class, the unknown characteristic (parameter) of interest is the proportion of the class members voting for a specific candidate, and the sample summary (statistic) is the observed proportion of voters favoring the candidate in a random sample of class members. Variation 2 leads students through a physical simulation for generating sample proportions by sampling, and re-sampling, marbles from a box.
Type: Problem-Solving Task
This task is intended as a classroom activity. Students pool the results of many repetitions of the random phenomenon (rolling dice) and compare their results to the theoretical expectation they develop by considering all possible outcomes of rolling two dice. This gives them a concrete example of what we mean by long term relative frequency.
Type: Problem-Solving Task
Tutorials
This video compares theoretical and experimantal probabilities and sources of possible discrepancy.
Type: Tutorial
Watch the video as it predicts the number of times a spinner will land on a given outcome.
Type: Tutorial
Parent Resources
Problem-Solving Tasks
The purpose of this task is to provide students with the opportunity to determine experimental probabilities by collecting data. The cylindrical objects used in this task typically have three different resting positions but not all of these may be equally likely and some may be extremely unlikely or impossible when the object is tossed. Furthermore, obtaining the probabilities of the outcomes is perhaps only possible through the use of long-run relative frequencies. This is because these cylinders do not have the same types of symmetries as objects that are often used as dice, such as cubes or tetrahedrons, where each outcome is equally likely.
Type: Problem-Solving Task
This task introduces the fundamental statistical ideas of using data summaries (statistics) from random samples to draw inferences (reasoned conclusions) about population characteristics (parameters). In the task built around an election poll scenario, the population is the entire seventh grade class, the unknown characteristic (parameter) of interest is the proportion of the class members voting for a specific candidate, and the sample summary (statistic) is the observed proportion of voters favoring the candidate in a random sample of class members. Variation 2 leads students through a physical simulation for generating sample proportions by sampling, and re-sampling, marbles from a box.
Type: Problem-Solving Task
This task is intended as a classroom activity. Students pool the results of many repetitions of the random phenomenon (rolling dice) and compare their results to the theoretical expectation they develop by considering all possible outcomes of rolling two dice. This gives them a concrete example of what we mean by long term relative frequency.
Type: Problem-Solving Task
