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L1-02 · Build a probability model

A probability model lists everything that could happen and how likely each one is. When every outcome is equally likely, the probability of something happening is the number of outcomes that count, out of all the outcomes. A probability goes from 0, which means impossible, to 1, which means certain.

Level 1 icon Level 1 · Grades 4–6 · about 40 minutes
Big idea. A probability model lists everything that could happen and how likely each one is. When every outcome is equally likely, the probability of something happening is the number of outcomes that count, out of all the outcomes. A probability goes from 0, which means impossible, to 1, which means certain.
Before this lesson. L1-01
Grown-ups. The guide for adults is at the end of this page.

What you will learn

Story: What are the chances?

Maya puts 8 counters in a paper bag. They are all the same size and feel the same. 3 are red and 5 are blue. “If I reach in without looking, what will I get?” she asks.

Leo makes a list of the outcomes. Each of the 8 counters is one outcome. Since they feel the same and Maya cannot see, each counter is equally likely to come out. 3 out of 8 outcomes are red. So their model says the probability of red is 3 out of 8. Leo writes it in short as P(red) = 3/8. The P stands for probability.

“What about not red?” asks Maya. Every counter that is not red is blue. That is 5 out of 8. Leo checks it a second way: 1 take away 3 eighths leaves 5 eighths. The two answers match, so P(not red) = 5/8.

Next they try a fair die. It has 6 faces, and each face is equally likely. The probability of rolling a 4 is 1 out of 6. The probability of not rolling a 4 is 5 out of 6.

“Can we roll a 7?” asks Leo. No face shows a 7, so the probability is 0. That is impossible. Rolling a number from 1 to 6 always happens, so its probability is 1. That is certain.

A bag of 8 counters becomes a probability model on a line from 0 to 1.

What the animation shows. Eight counters sit in a row: 3 red and 5 blue, and the red ones light up as 3 out of 8. The fraction P(red) = 3/8 slides onto a line from 0, marked impossible, to 1, marked certain. Then the blue ones light up as 5 out of 8. They fill the rest of the line, since P(not red) = 1 - 3/8 = 5/8.

Hands-on: Three probability models

About 20 minutes.

You need:

Steps:

  1. Put 2 counters of one color and 4 of another color in the bag. Write a list of the outcomes.

  2. Write the probability of each color as a fraction. Check that the two fractions add up to 1.

  3. Draw a circle and split it into 4 equal parts. Color 1 part yellow and 3 parts green. This is your spinner.

  4. Put the paper clip on the center and hold it with a pencil tip. Flick the clip to spin it.

  5. Write the probability of yellow and of green for your spinner.

  6. Write the probability of rolling a 6 on the die, and of not rolling a 6.

  7. Draw a long line. Write 0 and impossible at the left end. Write 1 and certain at the right end. Mark one half in the middle.

  8. Place every probability you found on the line. Add one event that is impossible and one that is certain.

Practice together

A bag has 1 green counter and 3 yellow counters, all the same size. You draw one without looking. What is the probability of green? What is the probability of not green?

Check your answer

There are 4 counters, and each is equally likely. 1 out of 4 is green, so the probability of green is one fourth. Not green is 3 out of 4. Check: 1 take away one fourth is three fourths.

Try it yourself

1. A spinner has 5 equal parts. 2 parts are orange and 3 are purple. What is the probability of orange? Where would you mark it on a line from 0 to 1?

Check your answer

The probability of orange is 2 out of 5, which is 0.4. It goes a bit to the left of one half on the line.

2. Leo draws a spinner with one big red part and one small blue part. He says red and blue each have a probability of one half, because there are two colors. Is he right?

Check your answer

No. The two parts are not the same size, so red and blue are not equally likely. The arrow will land on the big red part more often. Counting colors only works when each outcome is equally likely.

Talk about it

Check yourself

1. A bag has 3 red and 5 blue counters, all the same size. What is the probability of drawing blue?

Show the answer

C. 5 out of 8

There are 8 counters in all, and 5 of them are blue. So the probability of blue is 5 out of 8.

2. The probability of rain tomorrow is 0.3 in a weather model. What is the probability of no rain in that model?

Show the answer

A. 0.7

Rain and no rain together make 1. So no rain is 1 take away 0.3, which is 0.7.

3. Remember Lesson 1. A bag holds 12 marbles, and 3 of them are green. What fraction of the marbles are green? (Review of L1-01)

Show the answer

C. 1 out of 4

3 out of 12 marbles are green. That is the same as 1 out of 4.

Words to know

probability
A number from 0 to 1 that says how likely something is, according to a model.
outcome
One thing that could happen, like drawing one counter or rolling a 4.
model
A simple plan of how something works. A probability model lists the outcomes and how likely each one is.
For grown-ups: about this lesson

Why it matters. The lesson builds a probability model in the sense of the Common Core: probability as a number from 0 to 1 that expresses how likely an event is, with a probability near 0 indicating an unlikely event and one near 1 a likely event (7.SP.5), and a uniform model that gives equal probability to every outcome (7.SP.7) NGA Center and CCSSO, 2010. OpenStax Introductory Statistics defines outcomes, the sample space, equally likely outcomes, and the complement rule P(not A) = 1 - P(A) Barbara Illowsky and Susan Dean, 2023. Children see the complement two ways, by counting the other outcomes and by subtracting from 1, so the rule is checked rather than memorized.

Watch for these mix-ups.

  • Two possible results always means a 50-50 chance. (Only if the two results are equally likely, as the unequal spinner shows.)

  • A probability of 3 out of 8 means exactly 3 reds in every 8 draws. (It is a prediction from a model; real counts vary, as Lesson 3 shows.)

  • Probabilities can be bigger than 1 or less than 0. (Every probability is from 0 to 1.)

  • A very unlikely event is impossible. (Impossible means probability 0. A small probability can still happen.)

To make it easier. Use a bag with 1 red and 1 blue counter first, then 1 red and 3 blue, and write only fractions with 2 or 4 on the bottom.

To make it harder. Add a third color to the bag and check that the three probabilities add up to 1. Or draw a spinner with parts of 1 half, 1 fourth, and 1 fourth, and ask why counting colors would give the wrong model.

Limits. The model assumes each counter or face is equally likely. Real draws can be slightly biased, for example if one counter is bigger, stickier, or always on top. The lesson does not claim that real draws will match the model exactly.

On the computer. Grown-ups can see where this idea leads later in Lab: Chance and repeated trials, written for older learners.

Sources for this guide. NGA Center and CCSSO, 2010Barbara Illowsky and Susan Dean, 2023

Next: L1-03 · Repeat and record

References
  1. NGA Center and CCSSO. (2010). Common Core State Standards for Mathematics. Common Core State Standards Initiative. https://learning.ccsso.org/wp-content/uploads/2022/11/Math_Standards1.pdf
  2. Barbara Illowsky and Susan Dean. (2023). 3.1 Terminology. OpenStax, Introductory Statistics 2e. https://openstax.org/books/introductory-statistics-2e/pages/3-1-terminology