How likely, from 0 to 1. What should happen and what did; why more trials bring the two together; lists, tables and tree diagrams for two things at once; and how a fair handful — a random sample — lets you say something honest about the whole bag. Study cards, hints, a practice quiz at three levels, and four workshop activities that check themselves.
That is probability: the part divided by the whole, written as a number from 0 to 1. 3 blue out of 10 marbles is 3/10, or 0.3, or 30%.
Four things people get wrong
⚠️People often think…
Three colors on a spinner means each one has a ⅓ chance.
Only if the sections are the same size. Three colors on a spinner that is half red is not ⅓ red — count equal sections, not colors.
Count equal chances, not names.
⚠️People often think…
5 blue out of 10 marbles is a probability of 5/5.
The bottom number is the WHOLE bag, not the ones left over. 5 blue out of 10 marbles is 5/10.
Part over whole. Never part over the rest.
⚠️People often think…
Two coins have three outcomes: two heads, two tails, or one of each.
There are four: HH, HT, TH, TT. Heads-then-tails and tails-then-heads are two different outcomes, which is why “one of each” happens twice as often.
List them out. Order counts.
⚠️People often think…
After five heads in a row, tails is due.
The coin has no memory. The next flip is still ½, exactly like the first one. Nothing is owed and nothing is due.
Past flips change nothing.
Part over whole, every time
1Watch one
A bag has 4 red marbles and 6 blue. What is the probability of red?
The part is the reds: 4.
The whole is every marble: 10.
4 out of 10.
P(red) = 4/10 = 0.4.
2Do one with me
A spinner has 8 equal sections. 3 of them are green. What is the probability of green?
The part is
The whole is
So the probability is
💬One sentence, then you move on
Why is the bottom number 8 and not 5?
3Try one
You flip two coins. How many different outcomes are there in all?
I want a hint first
Write the first coin, then the second: H-H, H-T, T-H, T-T. Heads-then-tails is not the same as tails-then-heads.
💬Last one — then you're done here
Why doesn't a coin remember the last five flips?
Where this goes
Where this lives
A 30% chance of rain, a free-throw percentage, a poll of 900 people standing in for a whole state. How the sample was chosen matters more than how big it is.
What this feeds
This is the last unit of seventh grade. Eighth grade opens with exponents, roots and scientific notation.
Name one thing from your day whose chance was somewhere between 0 and 1.
One card at a time — tap “Show me” to check yourself, then Next. Start at Foundation; when those feel easy, climb.
Helpful Hints
🪜 From a guess to a number — the ladder
List the outcomes (sample space) → circle the ones you want (event) → favorable ÷ all, IF they are equally likely → or run the experiment: happened ÷ trials → multiply by the number of tries to predict → more trials, closer match.
List, circle, divide, predict, repeat.
🤝 The whole topic in one idea
Probability and sampling are the same idea pointed in opposite directions. Probability starts with what you KNOW about the whole bag and predicts the handful. Sampling starts with the handful and estimates the whole bag. Both only work when the handful is drawn fairly.
⚠️ Traps the test loves
Three colors does not mean ⅓ each. Check that the outcomes are equally likely.
Probability is part ÷ WHOLE, not part ÷ the rest. 5 blue of 10 marbles is 5/10, not 5/5.
HT and TH are two different outcomes. Two coins have four outcomes, not three.
The coin has no memory. After five heads, the next flip is still ½.
A big sample can still be biased. HOW it was chosen matters more than how many.
📐 Two kinds of probability, two kinds of sample — the chart
Name
Where it comes from
Example
Theoretical probability
reasoning about equally likely outcomes
P(even) on a number cube = 3/6
Experimental probability
data: happened ÷ trials
a cup landed on its side 28 of 40 tosses
Random sample
everyone has an equal chance to be picked
a computer draws 40 student IDs
Biased sample
picked by convenience, by volunteers, or in one place
asking the soccer team about favorite sports
🎯 How the test will ask
A bag or a spinner — “what is P(…)?” (favorable ÷ all; try the complement for “not”).
A results table — “what is the experimental probability?” (count ÷ total trials).
“About how many in 300 tries?” — probability × 300.
Two stages — make a list, a table or a tree, then count paths.
A survey plan — “is this sample random or biased? why?”
Sample results and a population size — scale up with a proportion, and say “about.”
✅ Before the test, can you…
Place an event on the line from 0 (impossible) to 1 (certain)?
Find a theoretical probability and an experimental one, and say why they differ?
Predict how many times something will happen in many trials?
Build the sample space for a two-stage event with a tree or a table?
Tell a random sample from a biased one and explain the bias?
Use a sample to estimate a population — and explain why another sample would differ?
Pick your level
Look back at anything you missed — the hint that appeared is exactly what to reread tonight.
How sure did you feel?
Workshop
Work like a statistician: sort by how likely, graph what happened against what should, finish a tree diagram, then judge whether a survey can be trusted. Every activity checks itself, and hints are free.
Your practice never leaves this device. There is no account and no sign-in. Your work is saved in this browser only, and you can erase it whenever you want.
Your practice record — saved on this device
This is your record of the module on screen — it stays here and goes nowhere. Independent means you got it right on the first tap; supported means you got it after the explain-and-retry, or marked ‘I had it’ on a revealed answer. Both count, and neither is a grade. If your teacher asks, copy the row or show them this screen.
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Supported
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Lab
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Traps met
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Explained in own words
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The answer key is for a teacher: it prints only from here, for the unit on screen. Print the study packet prints the study pages and a blank quiz — never the answers.