7.SP.C.7Common CoreMathStatistics and ProbabilityGrade 7
7.SP.C.7: Building and Testing Probability Models
In plain English: 7.SP.C.7 is the Common Core grade 7 math standard that asks students to build a probability model, a list of the possible outcomes with a probability for each, and use it to find probabilities of events. Models can be uniform, with equally likely outcomes, or built from observed frequencies. Students compare a model's predictions with real data and explain why the two might disagree.
Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
a.Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.
b.Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?
Common Core State Standards for Mathematics · Domain: Statistics and Probability (SP) · Cluster: Investigate chance processes and develop, use, and evaluate probability models. Also written as 7.SP.7 · Official standard
Students build probability models. A probability model lists every possible outcome of a chance process with a probability for each one. The probabilities are between 0 and 1, and together they add to 1. Students build models in two ways. In a uniform model every outcome gets the same probability, as when a student is picked at random from a class (part a). In a model built from data, each probability is the relative frequency observed in many trials, as when a paper cup is tossed 100 times (part b). Such a model is often not uniform.
Then students test models against data. They use a model to predict counts, compare the predictions with observed frequencies, and decide whether the agreement is good. When it is not, they explain possible reasons: too few trials, a spinner that sticks, a coin that is not balanced, or a bag that was not mixed. All data sets on this page are invented for teaching and labeled that way.
Learning Objectives
By the end of this lesson, students will be able to:
List the outcomes of a chance process and write a uniform probability model for it
Use a model to find the probability of an event, such as picking a girl from a class
Build a probability model from observed frequencies and decide whether it looks uniform
Compare a model's predictions with observed frequencies and judge whether they agree well
Explain possible sources of a discrepancy between a model and data
Prior Knowledge Required
Students should already be comfortable with:
Probability as a number from 0 to 1 7.SP.C.5
Relative frequency, the frequency of an event divided by the number of trials 7.SP.C.6
Simplifying fractions and converting them to decimals, for example 3/8 = 0.375 7.NS.A.2
Ratio language, such as "3 red cubes for every 1 blue cube" 6.RP.A.1
"A bag holds 3 red cubes and 1 blue cube. Jonah says: 'There are two colors, so the probability of red is 1/2.' Is he right? Explain."
Let students argue for a minute, then draw the four cubes on the board as R, R, R, B. Each cube is equally likely to be drawn, so the probability of red is 3/4, not 1/2. The lesson idea: equal chances belong to the four cubes, not to the two colors. Tell students they will learn to decide which outcomes are equally likely, and what to do when none are.
Direct Instruction20 minutes
The sample space is the list of all possible outcomes. For one roll of a number cube (a cube with the numbers 1 to 6 on its faces), it is 1, 2, 3, 4, 5, 6.
A probability model gives each outcome in the sample space a probability. Each probability is between 0 and 1, and the probabilities add to 1.
In a uniform model, every outcome has the same probability. With n equally likely outcomes, each has probability 1/n, and P(event) = (number of outcomes in the event) ÷ n. Picking at random means every outcome has the same chance.
When outcomes are not equally likely, build the model from data: toss or spin many times and use each outcome's relative frequency as its probability. This model may be non-uniform (not uniform).
To test a model, multiply each probability by the number of trials to get the predicted counts, then compare them with the observed frequencies (the counts that really happened). A large difference is a discrepancy, and it needs an explanation.
Uniform model, the official example (a)
A class has 28 students: 15 girls and 13 boys. Jane is one of the girls. The teacher picks one student at random.
Equation: Uniform model: each student has probability 1/28. P(Jane) = 1/28. P(a girl) = 15/28, about 0.54, because 15 of the 28 equally likely outcomes are girls.
Uniform model over cards, not letters
The letters of BANANA are written on 6 cards, and one card is drawn at random. Find P(A), P(N) and P(B).
Equation: Each card has probability 1/6. There are 3 A cards, 2 N cards and 1 B card: P(A) = 3/6 = 1/2, P(N) = 2/6 = 1/3, P(B) = 1/6. The model for cards is uniform, but the model for letters is not. Check: 1/2 + 1/3 + 1/6 = 1.
Model from data, the official example (b)
A small paper cup was tossed 100 times. It landed on its side 71 times, open end down 20 times and open end up 9 times (invented data). Find the approximate probability that it lands open end down.
Equation: P(side) ≈ 71/100 = 0.71, P(open end down) ≈ 20/100 = 0.20, P(open end up) ≈ 9/100 = 0.09. The three add to 1. The outcomes are far from equally likely, so the model is not uniform (Diagram 1).
Are the outcomes equally likely? The spinning penny
A penny was spun on its edge on a table 200 times. It landed heads up 76 times and tails up 124 times (invented data). Do heads and tails appear equally likely?
Equation: P(heads) ≈ 76/200 = 0.38 and P(tails) ≈ 124/200 = 0.62. A uniform model predicts 100 of each. A gap of 24 in 200 spins is large, so in this data the outcomes do not appear equally likely. Spinning is a different chance process from flipping, so a class should test its own pennies.
Compare a model with data, and explain a discrepancy
A spinner has 3 equal sections, A, B and C. In 90 spins it landed on A 41 times, B 25 times and C 24 times (invented data). Does the uniform model agree with the data?
Equation: Model: P = 1/3 for each section, so the prediction is (1/3)(90) = 30 each. A came up 41 times, 11 more than predicted. Possible sources: the spinner sticks or is bent, the table is tilted, students gave weak spins, or it is just chance in 90 spins. Check the spinner, then spin it many more times (Diagram 2).
For the official example (a), ask: "Why is P(a girl) bigger than P(Jane)?" (15 outcomes are girls, and only 1 outcome is Jane.) For the paper cup, ask: "Could we have found these probabilities without tossing?" (No: the cup's shape makes the outcomes unequal, and only data shows how unequal.) Point out that a model built from data is an estimate, and it gets better with more trials.
Guided Practice15 minutes
Pairs solve these four problems. For each one, they first say whether the model is uniform or built from data.
Guided practice problems with answers
Problem
Answer
A bag has 7 green, 3 yellow and 2 purple marbles. One marble is drawn at random. Find P(green) and P(not purple).
Uniform over the 12 marbles: P(green) = 7/12, P(not purple) = 10/12 = 5/6
The letters of PROBABILITY are on 11 cards. One card is drawn at random. Find P(B) and P(a letter that comes before M in the alphabet).
P(B) = 2/11. Before M: A, B, B, I, I, L, so P = 6/11
A spinner with unequal sections landed on A 21 times, B 12 times and C 17 times in 50 spins (invented data). Build a model from the data.
P(A) ≈ 0.42, P(B) ≈ 0.24, P(C) ≈ 0.34; they add to 1
A fair number cube (one whose faces are equally likely) is rolled 120 times, and a 1 comes up 26 times. What does the uniform model predict? Is the agreement reasonable?
(1/6)(120) = 20. 26 is 6 more, a difference chance can easily cause in 120 rolls, so the agreement is reasonable
Listen for students who treat colors or letters as the equally likely outcomes, as Jonah did in the warm-up. Ask: "What are the objects that each have the same chance?"
Independent Practice10 minutes
Independent practice problems with answers
Problem
Answer
In a class of 30, 12 students walk to school, 9 bike and 9 ride the bus. One student is picked at random. Find P(bikes) and P(does not walk).
A number cube is rolled once. Find the probability of rolling a factor of 6 (a number that divides 6 with no remainder).
1, 2, 3 and 6: 4/6 = 2/3
A rubber eraser shaped like a box was tossed 200 times. It landed on a large face 142 times, a long side 48 times and a small end 10 times (invented data). Build a model.
A coin was flipped 40 times and landed heads 29 times. What does a uniform model predict? Is the agreement good? Give one possible reason.
20 heads. No, 29 is far from 20. Possible reasons: the flips were not high or random enough, the coin is a trick coin, or it was chance in only 40 flips
Closure5 minutes
Exit ticket: A bag has 4 red and 6 blue cubes. (1) Write the uniform model for drawing one cube, and find P(blue). (Answer: each cube 1/10; P(blue) = 6/10 = 0.6.) (2) In 50 draws, putting the cube back each time, blue came up 41 times. What does the model predict? (Answer: 0.6 × 50 = 30.) (3) Give one possible reason for the discrepancy.
Differentiation Strategies
For Struggling Students
Have students write out the whole sample space first, one outcome per line, before writing any probability
Use a two-column model table (Outcome, Probability) and a third row that checks the sum is 1
Start with bags of cubes students can see and count, then move to spinners and cups
For Advanced Students
Ask students to design a spinner whose model is P(red) = 1/2, P(blue) = 1/3, P(green) = 1/6, and to give the angle of each section
Give data from 20 spins and from 2,000 spins that disagree with a model by the same percent, and ask which is stronger evidence against the model
Ask whether the paper cup model would change for a larger cup, and how to test it
Assessment Guidance
What to Look For
Check that every model lists all outcomes and that its probabilities add to 1. For uniform models, students should name the equally likely objects (students, cards, faces), not categories. When comparing a model with data, a strong answer computes the predicted counts first, says how far each observed count is from its prediction, and gives a physical reason (a sticking spinner, an unmixed bag) as well as chance with few trials.
02
Classroom Activities
3 Activities
1
Paper Cup Toss
15 minGroups of 3
Groups carry out the official example (b): they toss a small paper cup, build a model from the data, and compare it with a uniform model.
Procedure
Before tossing, each student guesses the probability of each outcome: on its side, open end down, open end up
One student tosses the cup gently to a height of about 50 cm above the desk, one names the outcome, and one tallies. Rotate jobs every 10 tosses
Each group makes 50 tosses and writes its model: each outcome's frequency divided by 50
The class adds all groups' tallies and writes a class model
Discussion Questions
Is the cup's model uniform? How do you know?
How close were your guesses to the class model?
Why might two groups get different models from the same kind of cup?
If the class tossed a larger cup, would you use the same model? How would you check?
Modification for Distance Learning
Students toss any cup at home 50 times and enter their tallies in a shared table, which adds them into a class model.
2
Spin It or Flip It?
15 minPairs
Pairs test whether heads and tails are equally likely for two different chance processes with the same penny: spinning it on its edge and flipping it.
Procedure
Spin the penny on its edge on a smooth desk 40 times and record heads up or tails up. Spins that fall off the desk do not count
Flip the same penny 40 times, catching it in your hand, and record the results
For each process, compare the counts with the uniform model's prediction of 20 heads and 20 tails
Pool the class data for each process and find the relative frequency of heads
Discussion Questions
Based on the class frequencies, do the spinning outcomes appear to be equally likely? Do the flipping outcomes?
A pair got 25 heads in 40 spins. Is that enough to say spinning favors heads? Why or why not?
What could make spin results differ from flip results, even with the same penny?
Challenge Variation
Repeat the spins with a different coin, such as a nickel or a quarter. Does the same model fit both coins?
3
Fair Spinner, Tricky Spinner
15 minGroups of 3-4
Each group gets two spinner templates. Spinner 1 has 4 equal sections. Spinner 2 has 4 sections that look equal but are not: one is 120° and the other three are 80° each. Groups test both spinners against a uniform model before they measure anything.
Procedure
Make a spinner: hold a paper clip at the center with a pencil point and flick the clip
Spin each spinner 60 times and tally the sections. The uniform model predicts (1/4)(60) = 15 for each section
Compare the counts with 15. For the spinner that disagrees more, list possible reasons
Now measure the angles with a protractor and write the true model for Spinner 2: 120/360 = 1/3 for the large section and 80/360 = 2/9 for each other section
Discussion Questions
For Spinner 2, the true model predicts (1/3)(60) = 20 spins for the large section. Were your counts closer to 15 or to 20?
Which reasons on your list turned out to be true? Which were chance?
How many spins would help you tell a 1/4 section from a 1/3 section with more confidence?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: A Model Built from Paper Cup Data
Invented data for 100 tosses of a small paper cup. Each bar is an outcome's relative frequency, which becomes its probability in the model: 0.71, 0.20 and 0.09, adding to 1. The dashed line shows the 1/3 a uniform model would give each outcome. The data are far from it, so the model is not uniform. Drawn to scale.
Diagram 2: Comparing a Uniform Model with Spinner Data
A spinner with 3 equal 120° sections, and invented data for 90 spins. The dashed bars show the uniform model's prediction, (1/3)(90) = 30 for each section. The solid bars show the observed counts: 41, 25 and 24. Section A is 11 above the prediction, so the agreement is not good, and students look for a cause. Drawn to scale.
04
Homework Assignment
~30 min
7.SP.C.7 Homework: Probability Models and Real Data
Directions: Show your work. For every model, list all the outcomes and check that the probabilities add to 1. When you compare a model with data, give the predicted counts first.
Part 1: Uniform Models (Problems 1-2)
A class has 32 students: 18 are in the school band and 14 are not. Diego is in the band. The teacher picks one student at random to lead the warm-up. Find (a) P(Diego), (b) P(a band member), (c) P(not a band member). Write each answer as a fraction in lowest terms.
A spinner has 12 equal sections numbered 1 to 12. (a) Write the probability model. (b) Find P(a multiple of 3). (c) Find P(a number less than 6). (d) Find P(13), and explain your answer.
Part 2: Models from Data (Problems 3-4)
A sneaker was dropped from waist height 120 times. It landed right side up 67 times, on its side 43 times and upside down 10 times (invented data). (a) Build a probability model from the data, with decimals to the nearest hundredth. (b) Is the model uniform? Explain. (c) Predict the number of upside-down landings in 300 more drops.
Sam spun a coin on its edge 80 times and got 31 heads and 49 tails. Luis flipped a coin 80 times and got 42 heads and 38 tails (invented data). (a) Find the relative frequency of heads for each. (b) Whose results look more like equally likely outcomes? (c) What should Sam do to be more sure about his coin?
Part 3: Comparing Models with Data (Problems 5-6)
A bag has 3 red, 5 green and 2 white cubes. Students drew a cube 200 times, putting it back each time: red 58, green 104, white 38 (invented data). (a) Write the model. (b) Find the predicted count of each color. (c) Does the model agree well with the data? Explain.
A spinner looks like it has 4 equal sections, A, B, C and D. In 200 spins it landed on A 71 times, B 44 times, C 42 times and D 43 times (invented data). (a) What does the uniform model predict for each section? (b) Is the agreement good? (c) Give two possible sources of the discrepancy. (d) Build a model from the data.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Building Models
All outcomes listed, probabilities correct and adding to 1
Model mostly correct, or the sum is not checked
Outcomes missing or probabilities wrong
Finding Probabilities
Event probabilities correct, counting the equally likely objects
One counting error
Counts categories instead of objects, or most answers wrong
Comparing with Data
Predicted counts found and compared with each observed count
Comparison made without predicted counts
No comparison
Explaining Discrepancies
Gives specific physical reasons and mentions chance with few trials
Gives one general reason
No reason given
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, 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
A class has 25 students, and 14 of them are girls. Priya is one of the girls. The teacher picks one student at random. What is the probability that Priya is picked?
Answer: C
Each of the 25 students is equally likely, so P(Priya) = 1/25. Choice A divides by the 14 girls only, as if the teacher picked from the girls. Choice B is the probability of picking a girl. Choice D divides by the 11 boys.
Question 2 of 20 · Multiple Choice
In the same class of 25 students with 14 girls, what is the probability that the student picked is a boy?
Answer: D
There are 25 - 14 = 11 boys, so P(boy) = 11/25. Choice A compares boys with girls instead of with the whole class. Choice B gives the chance of one particular boy among the boys. Choice C is the probability of a girl.
Question 3 of 20 · Multiple Choice
Which of these is a uniform probability model?
Answer: D
A uniform model gives every outcome the same probability, and only the number cube does: 1/6 for each face. Choices A, B and C are all valid models, since their probabilities add to 1, but their outcomes have different probabilities.
Question 4 of 20 · Multiple Choice
A spinner has 10 equal sections numbered 1 to 10. What is the probability of spinning a number greater than 7?
Answer: B
8, 9 and 10 are greater than 7: 3 of the 10 equally likely outcomes, so 3/10. Choice A counts the numbers from 1 to 7 instead. Choice C counts 7 as greater than 7. Choice D treats "greater than 7" as a single outcome.
Question 5 of 20 · Multiple Choice
A probability model for a spinner gives P(red) = 0.25 and P(blue) = 0.40. Green is the only other outcome. What is P(green)?
Answer: C
The probabilities must add to 1: 1 - 0.25 - 0.40 = 0.35. Choice A is P(red) + P(blue), not what is left. Choice B assumes the three colors are equally likely. Choice D subtracts 0.25 from 0.40.
Question 6 of 20 · Multiple Choice
A toy car was pushed off a low step 200 times. It landed on its wheels 88 times, on its side 96 times and on its roof 16 times (invented data). What probability does a model built from this data give for landing on its side?
Answer: B
Divide the frequency by the number of trials: 96/200 = 0.48. Choice A divides 96 by 100 instead of by 200. Choice C assumes a uniform model with three equally likely outcomes. Choice D is the probability of landing on its roof, 16/200.
Question 7 of 20 · Multiple Choice
A spinner with 4 equal sections was spun 80 times: red 22, blue 19, green 18, yellow 21 (invented data). How do the results compare with the uniform model?
Answer: A
The model predicts (1/4)(80) = 20 for each color, and every count is within 2 of 20, which is normal for chance. Choice B treats a difference of 2 as large. Choice C divides 80 by 4 incorrectly. Choice D reads small chance differences as proof.
Question 8 of 20 · Multiple Choice
A spinner with 3 equal sections was spun 180 times. The uniform model predicts 60 for each section, but the counts were 91, 45 and 44 (invented data). Which is the most reasonable explanation?
Answer: C
A count of 91 when 60 is predicted is a large difference in 180 spins, so something about the spinner, such as sticking or bending, may make one section more likely. Choice A is too strong: 180 spins can show a difference this large. Choice B ignores the 31-spin gap. Choice D is false: a spinner has no memory.
Question 9 of 20 · Multiple Choice
A penny was spun on its edge 60 times. It landed heads up 22 times and tails up 38 times (invented data). Based on these frequencies, what is the best estimate of P(tails)?
Answer: D
38/60 ≈ 0.63. Choice A assumes heads and tails are equally likely without looking at the data. Choice B is the estimate for heads, 22/60. Choice C divides 38 by 22, tails compared with heads, and gives a number greater than 1, which cannot be a probability.
Question 10 of 20 · Multiple Choice
The letters of MISSISSIPPI are written on 11 cards, and one card is drawn at random. What is P(S)?
Answer: B
4 of the 11 equally likely cards show S, so P(S) = 4/11. Choice A treats the 4 different letters (M, I, S, P) as equally likely. Choice C gives the chance of one particular S card. Choice D compares the 4 S cards with the 7 other cards.
Question 11 of 20 · Multiple Choice
A bag has 6 red, 4 blue and 2 yellow cubes. One cube is drawn at random. What is P(yellow)?
Answer: A
2 of the 12 cubes are yellow: 2/12 = 1/6. Choice B divides by the 10 cubes that are not yellow. Choice C treats the three colors as equally likely. Choice D counts the 2 yellow cubes as one outcome.
Question 12 of 20 · Multiple Choice
A student rolls a number cube 30 times and gets eight 6s. A uniform model predicts 5. What is the best conclusion?
Answer: C
In only 30 rolls, 3 more 6s than predicted can easily happen by chance, so the fair way to decide is to collect more data. Choices A and B read a small difference as proof. Choice D is false: a number cube has no memory of earlier rolls.
Question 13 of 20 · Multiple Choice
Which data set suggests that the outcomes are NOT equally likely?
Answer: A
Equally likely outcomes would give about 500 top up in 1,000 tosses, and 312 is far from 500. In choices B, C and D, each count is within a few of its prediction (100, 100 and 50), which chance easily explains.
Question 14 of 20 · Multiple Choice
In a raffle, each of 250 tickets is equally likely to be drawn. Tomas bought 5 tickets. What is the probability that one of his tickets is drawn?
Answer: B
5 of the 250 equally likely tickets are his: 5/250 = 1/50. Choice A is the probability for just one of his tickets. Choice C divides 1 by the 5 tickets he bought. Choice D divides his 5 tickets by the other 245 tickets: 5/245 = 1/49.
Question 15 of 20 · Short Answer
A spinner has 8 equal sections labeled 1 to 8. (a) Write the uniform probability model. (b) Find P(an odd number) and P(a number less than 3).
(a) Each of the outcomes 1, 2, 3, 4, 5, 6, 7, 8 has probability 1/8. (b) Odd: 1, 3, 5, 7, so P = 4/8 = 1/2. Less than 3: 1 and 2, so P = 2/8 = 1/4.
Question 16 of 20 · Short Answer
A paper cone (like a party hat) was tossed 250 times. It landed on its side 185 times and on its open end 65 times (invented data). Build a probability model, and predict the number of side landings in the next 100 tosses.
P(side) ≈ 185/250 = 0.74 and P(open end) ≈ 65/250 = 0.26; they add to 1. Prediction: 0.74 × 100 = about 74 side landings.
Question 17 of 20 · Short Answer
A bag is supposed to hold 5 red and 5 blue cubes. In 100 draws, putting the cube back each time, red came up 78 times. (a) What does the model predict? (b) Is the agreement good? (c) Give two possible sources of the discrepancy.
(a) P(red) = 5/10, so about 50 reds. (b) No: 78 is 28 more than predicted, far more than chance usually gives in 100 draws. (c) For example: the bag really holds more red cubes than claimed, the cubes were not mixed between draws, a red cube is larger and easier to grab, or someone peeked.
Question 18 of 20 · Short Answer
A penny was spun on its edge 100 times and landed tails up 64 times (invented data). Do heads and tails appear equally likely? Explain using the frequencies.
A uniform model predicts 50 tails, and the data show 64, so P(tails) ≈ 0.64 and P(heads) ≈ 0.36. A gap of 14 in 100 spins is large, so for this penny the outcomes do not appear equally likely. More spins would make the conclusion stronger.
Question 19 of 20 · Short Answer
A class has 30 students, and 16 are girls. Rosa is one of the girls. (a) One student is picked at random from the whole class. Find P(Rosa) and P(a girl). (b) If instead one girl is picked at random from the girls only, what is P(Rosa)?
(a) P(Rosa) = 1/30 and P(a girl) = 16/30 = 8/15. (b) There are now 16 equally likely outcomes, so P(Rosa) = 1/16.
Question 20 of 20 · Short Answer
Explain the difference between a uniform probability model and a non-uniform one. Then say which kind of model fits each situation, and give its probabilities: (a) one name is drawn from a well-mixed hat holding the names of 24 students; (b) one card is drawn at random from 5 cards that spell TEETH, and you record the letter.
In a uniform model every outcome has the same probability; in a non-uniform model the outcomes have different probabilities. (a) Uniform: each of the 24 names has probability 1/24. (b) Non-uniform for letters: the 5 cards are equally likely, but P(T) = 2/5, P(E) = 2/5 and P(H) = 1/5. Check: 2/5 + 2/5 + 1/5 = 1.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 7.SP.C.7 mean?
7.SP.C.7 means students build probability models, use them to find probabilities, and check them against real data. Part a covers uniform models, where every outcome is equally likely, like picking a student at random. Part b covers models built from observed frequencies, like tossing a paper cup many times. When a model and the data disagree, students explain why.
What is a probability model in 7th grade math?
A probability model is a list of all the possible outcomes of a chance process with a probability for each one. The probabilities are between 0 and 1 and add to 1. For a fair number cube, the model gives each of the faces 1 to 6 a probability of 1/6.
What is the difference between a uniform and a non-uniform probability model?
In a uniform model all outcomes have the same probability, and in a non-uniform model they do not. Drawing one card from a well-mixed deck of cards numbered 1 to 20 is uniform. A tossed paper cup, which lands on its side much more often than on its base, needs a non-uniform model built from data.
What grade is 7.SP.C.7, and what comes next?
7.SP.C.7 is a grade 7 standard in Statistics and Probability. It builds on relative frequency from 7.SP.C.6 and leads to compound events, such as rolling two number cubes, in 7.SP.C.8. In high school, students describe events as subsets of a sample space (HSS.CP.A.1) and decide whether a model is consistent with data (HSS.IC.A.2).
Why don't observed frequencies match the model exactly?
Chance results vary, so observed counts are almost never exactly equal to the predicted counts. Small differences are normal, especially with few trials. A large difference may point to a problem: a spinner that sticks, a bag that was not mixed, an object that is not balanced, or a recording mistake. The standard asks students to name such possible sources.
How do students decide whether the agreement is good?
In grade 7 the decision is a judgment, not a formal test. Students compare each observed count with its predicted count and ask whether the gap is small compared with the number of trials. A gap of 4 heads in 60 coin flips is easy to get by chance; a gap of 30 in 120 draws is not. Collecting more data is always a fair next step.
Does a spinning penny land heads up half the time?
Students should not assume it; the official example asks them to decide from data. Spinning a penny on its edge is a different chance process from flipping it, and the results can depend on the coin and the surface. Have the class spin pennies many times, pool the results and judge whether heads and tails appear equally likely.
What does "selected at random" mean in 7.SP.C.7?
Selected at random means every member of the group has the same chance of being picked. Drawing names written on identical slips from a well-mixed bag, or using a random number generator with each student numbered, are two fair ways. Picking the first student who raises a hand is not random.
What mistakes should teachers watch for?
A common mistake is treating categories as equally likely outcomes, for example saying P(red) = 1/2 because a bag has two colors. Other frequent errors are models whose probabilities do not add to 1, dividing by the wrong total, and calling a small chance difference proof that a model is wrong.
How can parents help with 7.SP.C.7 at home?
Parents can ask their child to predict and then test. Put 3 spoons and 1 fork in a bag, and ask for the probability of drawing a spoon (3/4). Then draw, record and put back 40 times, and compare the count with the prediction of 30. Talk about why the result is close but not exact.
07
Related Standards
6 standards
These standards connect to 7.SP.C.7: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
7.SP.C.6Prerequisite
Estimate probability from long-run relative frequency and predict frequencies
