HSS.IC.A.2Common CoreMathStatistics and ProbabilityGrades 9-12
HSS.IC.A.2: Testing a Model Against Data with Simulation
In plain English: HSS.IC.A.2 is the Common Core statistics standard that asks students to decide whether a stated probability model fits the results of a real chance process. Students simulate many outcomes under the model, see how often a result like the observed one occurs, and judge whether it is unusual enough to doubt the model. It is usually taught in Algebra II or an introductory Statistics course.
Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?
Common Core State Standards for Mathematics · Domain: Making Inferences and Justifying Conclusions (IC) · Cluster: Understand and evaluate random processes underlying statistical experiments Also written as HSS-IC.A.2 or S-IC.2 · Official standard
A probability model makes a claim about a chance process: a coin lands heads half the time, a spinner lands on red a quarter of the time, a player makes 80% of her free throws. Students learn to test such a claim against real results. The tool is simulation: generate many sets of outcomes as if the model were true, then ask how often the simulated results are at least as extreme as what actually happened.
If results like the observed one are common under the model, the data are consistent with it. If they almost never happen, the data give evidence against the model. Students also learn the limits of the method: consistent does not mean proven, and an unusual result is evidence, not certainty. The official example, 5 tails in a row from a coin said to land heads with probability 0.5, anchors the lesson.
Learning Objectives
By the end of this lesson, students will be able to:
Design a simulation that matches a stated probability model, defining one trial and the result to record
Use simulation results to estimate how often an outcome at least as extreme as the observed one occurs under the model
Decide whether a model is consistent with observed data and justify the decision in context
Explain why data consistent with a model do not prove it, and why an unusual result is evidence but not proof against it
Prior Knowledge Required
Students should already be comfortable with:
Developing a probability model from observed frequencies 7.SP.C.7
Designing and using a simulation to generate frequencies for compound events 7.SP.C.8
Approximating a probability by a long-run relative frequency 7.SP.C.6
Multiplying probabilities of independent events HSS.CP.A.2
Read the prompt aloud and ask students to vote with a show of hands before anyone explains:
Warm-Up Prompt
"A friend hands you a coin and says it is fair. You toss it 10 times. Would you doubt her if you got 6 heads? 8 heads? 10 heads? Where would you start to doubt, and why there?"
Record the votes. Students usually agree that 6 heads is not suspicious and 10 heads is, and they disagree about 8. Ask what would help them decide. Steer toward the idea that they need to know how often each result would happen if the coin really were fair. That question, "how often would this happen if the model were true?", is the whole lesson.
Direct Instruction20 minutes
Define a model as a stated set of probabilities for a chance process, and a data-generating process as the real process that produced the results (spinning a real coin, a real player shooting free throws). Present the simulation method:
State the model in probability terms, for example P(red) = 0.25.
Choose a random device that matches the model: random digits 0-9, a random integer generator or a spinner. With digits, P = 0.25 can be modeled by the two-digit numbers 00-24 out of 00-99.
Define one trial as one full repetition of the real process, with the same number of outcomes (40 spins if 40 spins were observed), and name the result to record.
Run many trials, 100 or more when a computer is available, and record the result of each.
Compare: find the fraction of trials whose result is at least as extreme as the observed one.
Decide in context: if that fraction is small, the data are unusual under the model and give evidence against it; otherwise the data are consistent with the model.
Explain that "small" is a judgment. Many classes use about 5% as a rough guideline for unusual, but students should treat it as a convention, not a rule, and should always say how strong the evidence is.
The official example (exact probability)
A model says a spinning coin falls heads up with probability 0.5. You spin it 5 times and get 5 tails. Would that cause you to question the model?
Equation: Under the model, P(5 tails) = (0.5)⁵ = 1/32 ≈ 0.031: about 3 of every 100 sets of 5 spins. That is unusual, so it gives some reason to question the model, but 5 spins are few; spin the coin many more times before rejecting it.
Simulation gives evidence against the model
A game spinner is said to land on red with probability 0.25. In 40 real spins it landed on red 16 times. In 200 simulated trials of 40 spins under the model (Diagram 1), 9 trials had 16 or more reds.
Equation: Estimated P(16 or more) ≈ 9/200 = 0.045. Results this extreme are unusual under the model, so the data give evidence that P(red) is greater than 0.25.
Simulation shows the data are consistent
A die is said to be fair, so P(six) = 1/6. In 60 rolls there were 14 sixes, when the model predicts about 10. In 500 simulated sets of 60 fair rolls, 54 had 14 or more sixes.
Equation: Estimated P(14 or more) ≈ 54/500 = 0.108. About 1 set in 9 reaches 14 or more, so 14 sixes is consistent with a fair die.
Deciding in context
A coach's model says a player makes 80% of her free throws. In one game she made 5 of 10. In 200 simulated games of 10 shots under the model, 6 had 5 or fewer makes.
Equation: Estimated P(5 or fewer) ≈ 6/200 = 0.03. The result is unusual under the model, so it raises doubt about the 80% figure for this player, though one bad game could also have a cause, such as an injury, that the model ignores.
Show Diagram 2 with the first example: the exact probability of each number of tails in 5 spins under the model. Point out that 5 tails is the least likely bar, tied with 5 heads. If you would have been just as suspicious of 5 heads, the chance of a result that extreme in either direction is 2/32 = 1/16 ≈ 0.06, which is less surprising. Show Diagram 1 with the spinner example and ask students to point to the observed value and the trials at least as extreme.
Guided Practice15-20 minutes
Work through one simulation as a class. A cereal company says 20% of its boxes contain a prize. A family opens 10 boxes and finds no prizes. Is the claim consistent with this result?
Model: P(prize) = 0.2. Random device: digits 0-9, where 0 and 1 mean prize and 2-9 mean no prize.
One trial: 10 random digits (10 boxes). Record the number of prizes.
Each student runs 5 trials from a random digit table, starting at a different row, and reports how many had 0 prizes. Pool the class results on the board.
Compare: the exact probability of no prize in 10 boxes under the model is 0.8¹⁰ ≈ 0.107, so the class should find about 1 trial in 9 or 10 with no prizes.
Decide: 0 prizes in 10 boxes happens often enough under the model that the result is consistent with the company's claim.
Listen for students who define one trial as a single digit instead of 10 digits, and for students who conclude "the claim is true" instead of "the data are consistent with the claim".
Independent Practice15 minutes
A town official's model says half of the town's voters favor a new recycling fee. In a random sample of 30 voters, 21 favored it. The table shows the results of 100 simulated samples of 30 voters under the model (computer simulation, P(favor) = 0.5).
Number in favor in 100 simulated samples of 30 voters, model P(favor) = 0.5
Number in favor
9
10
11
12
13
14
15
16
17
18
19
20
21
22
Number of trials
1
1
8
9
11
12
13
14
14
7
6
1
1
2
Tasks: (1) Explain what one trial represents. (2) How many trials had 21 or more in favor? (Answer: 3.) (3) Estimate the probability of a result at least this extreme under the model. (Answer: 3/100 = 0.03.) (4) Is the model consistent with the sample? Write two sentences in context. (5) Name one thing that could make the simulated table differ from a classmate's table.
Closure5-10 minutes
Exit ticket: (1) Complete the sentence: "A model is consistent with data when ...". (2) A simulation shows that a result at least as extreme as the observed one happened in 41 of 200 trials. What do you conclude about the model? (3) Why is it wrong to say a simulation "proves" a model is correct?
Differentiation Strategies
For Struggling Students
Give a simulation planning frame with four blanks: the model, the random device and what each outcome means, one trial, and the result to record
Start with a model that uses one digit per outcome (P = 0.5 as even and odd digits) before models such as P = 0.25 that need two-digit numbers
Have students circle the observed value on the simulation dot plot and shade every trial at least as extreme before computing any fraction
For Advanced Students
Ask students to compute the exact probability for the free-throw example with the binomial formula and compare it with the simulated estimate
Ask students to write a short program or spreadsheet that runs 1,000 trials and explain how the estimate changes compared with 100 trials
Ask students to explain why "at least as extreme" is used instead of "exactly equal to" the observed result, using a model with 100 outcomes
Assessment Guidance
What to Look For
Check that students define a trial as a full repetition of the real process with the same number of outcomes, and that their random device assigns the right probability to each outcome. Look for comparisons that count results at least as extreme as the observed one, not only results equal to it. Strong conclusions are stated in context and in measured language: "the data give evidence against the claim" or "the data are consistent with the claim", never "the claim is proven true" or "the model is definitely false".
02
Classroom Activities
3 Activities
1
Spin the Coin
20 minPairs
Pairs recreate the official example. They spin a real coin on a desk in sets of 5 spins, then simulate sets of 5 spins under the fair model, and compare how often each produces 5 of the same face.
Procedure
Each pair spins one coin (spun on its edge, not tossed) in 8 sets of 5 spins and records the number of tails in each set
Each pair then simulates 20 sets of 5 spins with random digits, using even digits for heads and odd digits for tails
Pool the class results in two tables: real sets and simulated sets, each tallied by number of tails from 0 to 5
Compare the class fraction of simulated sets with 5 tails to the exact value 1/32 ≈ 0.031, and with 5 of either face to 2/32 = 1/16
Discussion Questions
If a pair got 5 tails in one real set, should they conclude the coin is unfair? What would they need to see?
Does the real-spin table look like the simulated table? What could explain a difference?
How would you change the simulation to test a model that says P(heads) = 0.4?
Modification for Distance Learning
Students spin coins at home and enter their counts in a shared form; the simulation uses a random integer generator set to 0 or 1.
2
Is the Carnival Game Honest?
25 minGroups of 3-4
A carnival spinner has 6 equal sectors, one of which wins a prize, so the operator's model is P(win) = 1/6. A student reports that she played 50 times and won only twice. Groups design and run a simulation to decide whether her result is consistent with the operator's model.
Procedure
Groups agree on a random device: a random integer from 1 to 6, where 1 means a win, or a paper-clip spinner on a 6-sector circle
One trial is 50 plays. Record the number of wins
Each group runs 10 trials (with a calculator or online generator) and adds them to a class dot plot
Count the trials with 2 or fewer wins and estimate the probability of a result at least this extreme under the model
Groups write a conclusion in context and compare it with the teacher's exact value: under the model, P(2 or fewer wins in 50) ≈ 0.007
Discussion Questions
The model predicts about how many wins in 50 plays? (50/6 ≈ 8.3.)
If the class dot plot has no trial with 2 or fewer wins, does that mean the student's result was impossible under the model?
Besides a dishonest spinner, what else could explain the student's result?
Challenge Variation
Groups find the smallest number of wins in 50 plays that they would still call consistent with the model, using their simulated dot plot, and defend the choice to the class.
3
Model or Mismatch? Evidence Cards
20 minPairs
Each pair gets 6 cards. Every card states a model, an observed result and the summary of 200 computer-simulated trials under the model. Pairs decide whether each result is consistent with its model and rank the cards from strongest to weakest evidence against the model.
The 6 Cards
A student guessing on 4-choice questions is right with probability 0.25. She got 9 of 20 right. Simulation: 8 of 200 trials had 9 or more right
A factory says 10% of its phone cases are defective. A random 30 had 7 defective. Simulation: 5 of 200 trials had 7 or more
A traffic light is green 40% of the time when a driver arrives. A driver found it green on 11 of 20 arrivals. Simulation: 21 of 200 trials had 11 or more
A seed company says 90% of its seeds sprout. Only 38 of 50 sprouted. Simulation: 1 of 200 trials had 38 or fewer
A four-color spinner with equal sections landed on blue 3 times in 24 spins. Simulation: 23 of 200 trials had 3 or fewer blue
A coin is said to be fair. It landed heads 12 times in 15 tosses. Simulation: 4 of 200 trials had 12 or more heads
Procedure
For each card, compute the fraction of simulated trials at least as extreme as the observed result
Label the card "consistent" or "evidence against the model" and write one sentence in context
Rank the six cards and compare rankings with another pair
Discussion Questions
Which card gives the strongest evidence against its model? Which gives the weakest?
For a card labeled consistent, could the model still be wrong?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Simulated Results for the Red Spinner
Results of 200 computer-simulated trials of 40 spins under the model P(red) = 0.25, drawn to scale. Most trials give 7-13 reds, near the expected 10. The red bars show the 9 trials with 16 or more reds, the observed result or something more extreme: an estimated probability of 9/200 = 0.045.
Diagram 2: The Official Example, 5 Spins of a Coin
Exact probabilities of 0 to 5 tails in 5 spins when P(heads) = 0.5, drawn to scale. The counts 1, 5, 10, 10, 5, 1 out of 32 equally likely sequences give each bar. Five tails (red) has probability 1/32, about 0.031, so it is unusual under the model but not impossible.
04
Homework Assignment
~30 min
HSS.IC.A.2 Homework: Is the Model Consistent with the Data?
Directions: For each problem, state the model, say what one trial is when a simulation is used, find the fraction of results at least as extreme as the observed one, and write your conclusion in context. Use measured language: "consistent with" or "evidence against", not "proves".
Part 1: Exact Probabilities and Planning a Simulation (Problems 1-2)
A model says a spinner lands on blue with probability 0.25. (a) Under the model, find the probability of 4 blues in a row and the probability of no blue in 4 spins. (b) If you saw one of these results, which would make you question the model? Explain your reasoning.
A candy company says 30% of its candies are orange. A bag of 50 candies has only 8 orange ones. (a) Describe how to simulate one bag with random digits under the company's model. (b) In 200 simulated bags, 3 had 8 or fewer orange candies. Is the bag consistent with the company's claim? Explain.
Part 2: Using Simulation Results (Problems 3-4)
An airline says 90% of its flights arrive on time. In a random sample of 40 of its flights, 31 arrived on time. A computer ran 100 simulated samples of 40 flights under the airline's model. Number on time (number of trials): 31: 3, 32: 4, 33: 8, 34: 9, 35: 15, 36: 17, 37: 18, 38: 22, 39: 4. (a) How many trials had 31 or fewer on time? (b) Estimate the probability of a result at least this extreme. (c) Is the airline's claim consistent with the sample?
A student says she can tell tap water from bottled water. Her friends' model says she is just guessing, so each answer is right with probability 0.5. In a blind test she was right on 9 of 12 cups. In 100 simulated sets of 12 guesses, 7 had 9 or more right. Is her result consistent with guessing? What should her friends conclude, and what would give stronger evidence either way?
Part 3: Reasoning about Conclusions (Problems 5-6)
A coin landed heads 6 times in 10 tosses. Under the model P(heads) = 0.5, the probability of 6 or more heads is about 0.377. Under the model P(heads) = 0.6, the probability of 6 or fewer heads is about 0.618. (a) Is the result consistent with each model? (b) Explain why this shows that data consistent with a model cannot prove that the model is true.
A player's coach claims she makes 70% of her free throws. In one game she made 4 of 10. (a) Design a random-digit simulation of one game under the coach's model. (b) Run 20 trials by hand and record the number of makes in each. (c) Use your trials to decide whether 4 of 10 is consistent with the coach's claim, and compare your estimate with a classmate's.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Simulation Design
Random device matches the model and one trial repeats the full process
Device or trial partly correct
Missing or does not match the model
Using the Results
Counts results at least as extreme and computes the fraction correctly
Counts only equal results or one arithmetic error
No fraction or incorrect method
Conclusion in Context
Clear decision, stated in context with measured language
Decision correct but vague or overstated
No decision or unsupported
Reasoning about Evidence
Explains why consistent is not proof and unusual is not certainty
Partial explanation
Missing or incorrect
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Answer each question, then read the explanation. Your score updates as you go, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
What does it mean to say that observed data are consistent with a model?
Answer: B
Consistent means results like the observed one happen reasonably often under the model, so the data give no strong reason to doubt it. Choice A overstates: other models could fit the data too. Choice C is too strict, since chance results rarely match a prediction exactly.
Question 2 of 20 · Multiple Choice
A model says a coin lands heads with probability 0.5. What is the probability of 3 tails in a row under this model?
Answer: A
The spins are independent, so (1/2)(1/2)(1/2) = 1/8. Choice B is the probability of exactly 2 tails in 3 spins (or of exactly 1), not 3 in a row. Choice C multiplies 2 by 3 instead of raising 2 to the third power, and choice D forgets that all three spins must be tails.
Question 3 of 20 · Multiple Choice
You want to simulate a process in which each outcome is a success with probability 0.3, using random digits 0-9. Which assignment works?
Answer: C
Three of the ten equally likely digits give probability 3/10 = 0.3. Choice A uses four digits (0, 1, 2, 3), which models 0.4, a common off-by-one error. Choice B models 0.1 and choice D models 0.5.
Question 4 of 20 · Multiple Choice
In 200 simulated trials under a model, only 2 trials gave a result at least as extreme as the observed one. What is the best conclusion?
Answer: D
An estimated probability of 2/200 = 0.01 means such a result would rarely happen if the model were true, which is evidence against the model. Choice A reverses the meaning of a small fraction. Choice B assumes an error without a reason: rare results do show up in simulations.
Question 5 of 20 · Multiple Choice
In 200 simulated trials under a model, 58 trials gave a result at least as extreme as the observed one. What is the best conclusion?
Answer: A
58/200 = 0.29, so results like this happen in about 29% of trials under the model. That is not unusual, so the data are consistent with the model. Choice C overstates the conclusion: consistent data do not prove a model, since other models may fit as well.
Question 6 of 20 · Multiple Choice
A spinner's results are consistent with the model P(red) = 0.25. Which statement must be true?
Answer: C
Consistency only says the data do not contradict the model; values close to 0.25, such as 0.22 or 0.3, might fit the data equally well. Choices A and B treat consistency as proof. Choice D confuses a long-run probability with a guarantee about a few spins.
Question 7 of 20 · Multiple Choice
A model says a spinning coin lands heads with probability 0.5. What is the probability of 6 tails in a row under the model?
Answer: D
(1/2)⁶ = 1/64 ≈ 0.016, so 6 tails in a row would happen in fewer than 2 of every 100 sets of 6 spins under the model. Choice B is the probability for 5 in a row. Choice C multiplies 6 by 2 instead of raising 2 to the 6th power, and choice A multiplies the answer by 6.
Question 8 of 20 · Multiple Choice
Why should a simulation use many trials, such as 200, instead of 10?
Answer: B
The fraction of trials with an extreme result is an estimate, and it varies less from one simulation to another when there are more trials. Choice A confuses the simulation with the real data. Choices C and D are false: the model stays the same, and rare results stay rare.
Question 9 of 20 · Multiple Choice
Which is a correct way to simulate one roll of a fair six-sided die?
Answer: A
Each integer from 1 to 6 is equally likely, matching the model P(each face) = 1/6. Choice B includes impossible values 0, 7, 8 and 9. Choice D has seven equally likely values, so each would have probability 1/7. Choice C gives a count from 0 to 6 that is not equally likely.
Question 10 of 20 · Multiple Choice
A seed packet claims 80% of its seeds sprout. A gardener plants 20 seeds and 11 sprout. In 100 simulated plantings of 20 seeds under the claim, 1 had 11 or fewer sprout. What should the gardener conclude?
Answer: C
Only 1 trial out of 100 was this extreme, an estimate of about 0.01, so the result is very unusual under the claim: fewer than 80% of these seeds may sprout, or the growing conditions may have differed from those the claim assumes. Choice B overstates: strong evidence is not proof. Choice A ignores how rare the result is.
Question 11 of 20 · Multiple Choice
A student tosses a coin 12 times, gets 7 heads and says the coin is unfair. Under the model P(heads) = 0.5, the probability of 7 or more heads in 12 tosses is about 0.39. What is wrong with the student's reasoning?
Answer: D
An outcome that happens in about 39 of every 100 sets of 12 tosses is ordinary, not evidence of unfairness. Choice A treats any result above the expected value as suspicious. Choice B is false, and choice C would change nothing, since 5 tails is the same result.
Question 12 of 20 · Multiple Choice
A two-color spinner is said to land on red with probability 0.5. Which result in 20 spins would most make you question the model?
Answer: C
The model predicts about 10 reds. 17 is the farthest from 10, and results at least that extreme have probability of about 0.0013 under the model. Choices A and D are within 1 of the expected value, and choice B is a fairly common result.
Question 13 of 20 · Multiple Choice
A real process produced 25 outcomes. In a simulation to test a model for it, what should one trial be?
Answer: B
Each trial must repeat the whole real process under the model, with the same number of outcomes, so the simulated results can be compared with the observed one. Choice A is a common error: a single outcome cannot be compared with a result based on 25. Choice C uses the data instead of the model.
Question 14 of 20 · Multiple Choice
A company says 5% of its light bulbs are defective. A random sample of 100 bulbs has 5 defective bulbs. What is the best conclusion?
Answer: B
The model predicts 5% of 100 = 5 defective bulbs, exactly the observed number, so the result is typical under the model. Choice A still overstates: matching a prediction does not prove the claim. Choice C reverses the conclusion.
Question 15 of 20 · Short Answer
A school says 35% of its students ride the bus. In a random sample of 25 students, only 4 ride the bus. Describe step by step how to use random digits to test whether this result is consistent with the school's claim.
Use two-digit random numbers: let 00-34 mean "rides the bus" and 35-99 mean "does not", so P(bus) = 0.35. One trial is 25 two-digit numbers; record how many are 00-34. Run many trials (100 or more with technology). Find the fraction of trials with 4 or fewer bus riders. If that fraction is small, the sample gives evidence against the 35% claim. (Under the model, the exact probability is about 0.032, so a simulation should usually find that the result is unusual.)
Question 16 of 20 · Short Answer
Return to the official example: a spinning coin that the model says lands heads with probability 0.5 gives 5 tails in a row. A classmate spins it 5 more times and gets 5 more tails, so 10 tails in a row. How does this change your conclusion about the model?
Under the model, P(10 tails in a row) = (1/2)¹⁰ = 1/1024 ≈ 0.001. That is very unusual, far more than 5 in a row, so the data now give strong evidence against the model: the coin, when spun, probably does not land heads half the time. It is still not proof, but the model is hard to believe.
Question 17 of 20 · Short Answer
A video game says each treasure chest has a 1 in 10 chance of holding a rare item. A player opens 30 chests and gets no rare items. In 100 simulated sets of 30 chests under the game's model, 4 had no rare items. Is the game's claim consistent with the player's result?
The estimated probability is 4/100 = 0.04. The result is unusual under the model, so it gives some evidence against the 1-in-10 claim, but not strong evidence: about 4 of every 100 players would see this by chance. (The exact probability, 0.9³⁰, is about 0.042.) Opening more chests would give a clearer answer.
Question 18 of 20 · Short Answer
A student says: "The probability of the result was only 3% under the model, so the result was impossible, and the model is wrong." Explain what is right and what is wrong in this statement.
Right: a 3% result is unusual under the model, so it is evidence against the model. Wrong: it is not impossible. If the model is true, such results still happen about 3 times in 100, so the model could be correct and this could be one of those times. The conclusion should be "evidence against the model", not "the model is wrong".
Question 19 of 20 · Short Answer
In 250 simulated trials under a model, 12 trials gave a result at least as extreme as the observed one. Estimate the probability of such a result under the model and state a conclusion.
12/250 = 0.048. A result this extreme happens in about 5 of every 100 trials under the model, so it is unusual and gives moderate evidence against the model. Because 0.048 is close to the rough 5% guideline, a careful answer says the evidence is not strong and more data would help.
Question 20 of 20 · Short Answer
A die is said to be fair. In 30 rolls it showed a one 8 times, when the model predicts 5. In 200 simulated sets of 30 fair rolls, 25 had 8 or more ones. Is the result consistent with a fair die?
The estimated probability is 25/200 = 0.125, about 1 in 8. Results this extreme are not unusual for a fair die, so the data are consistent with the model. That does not prove the die is fair; it only means these 30 rolls give no strong reason to doubt it.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSS.IC.A.2 mean?
It means students decide whether a stated probability model fits the results of a real chance process. They compare the observed result with what the model would usually produce, often by simulating the model many times, and judge whether the result is unusual enough to doubt the model.
What is the answer to the 5 tails in a row example in HSS.IC.A.2?
Five tails in a row is unusual but not impossible under the model: its probability is (0.5)⁵ = 1/32, about 0.031. That gives some reason to question the model, but with only 5 spins a teacher should push students to collect more data before rejecting it. There is no single required answer; the standard asks students to reason about it.
Is there a cutoff for when a result is unusual?
No fixed rule is part of the standard. Many classes use about 5% as a rough guideline, but students should describe the strength of the evidence rather than apply a cutoff mechanically. A result with estimated probability 0.03 is weaker evidence than one with 0.002.
Why do we count results at least as extreme, not just the exact result?
Because any single exact result can be unlikely. With 100 coin tosses, even 50 heads exactly has a probability under 0.08. The useful question is how often the model produces something as far from its prediction as what happened, or farther.
How many trials should a simulation have?
Enough that the estimate is stable. Hand simulations of 20-50 trials per class are fine for building the idea, especially when pooled. With a calculator or computer, 200 to 1,000 trials give a more reliable estimate. The number of trials is different from the number of outcomes within one trial, which must match the real process.
What if the simulation and the exact probability do not match?
Some difference is expected, because a simulated estimate varies from one run to the next. With more trials the estimate usually moves closer to the exact value. A large, persistent gap usually means the random device does not match the model or a trial does not repeat the full process, so check both.
Is HSS.IC.A.2 taught in Algebra 2 or in Statistics?
It is usually taught in Algebra II or an introductory Statistics course, and in Math III in integrated pathways. It builds on grade 7 work with probability models and simulation (7.SP.C.7 and 7.SP.C.8).
Do students need the binomial formula for this standard?
No. The standard names simulation as a method, and simple cases like the coin example can use the product of probabilities. Students who know the binomial formula can check a simulation against the exact probability, as the advanced differentiation suggests, but it is not required.
How does this standard lead to significance testing?
It is the informal version of it. Later, in HSS.IC.B.5, students use simulation to decide whether a difference between two treatments is significant. In a Statistics course, the fraction of simulated results at least as extreme becomes the p-value.
What mistakes do students make with simulations?
Common ones: defining a trial as one outcome instead of a full repetition of the process, assigning digits that do not match the model's probability, running too few trials, and concluding that a model is "true" or "false" instead of consistent or not.
07
Related Standards
6 standards
These standards connect to HSS.IC.A.2: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
7.SP.C.7Prerequisite
Develop a probability model and use it to find probabilities of events
