Simple Probability & Complements

SECTION 1

Where Did Probability Come From?

Have you ever wondered why some things happen more often than others? People have asked this question for thousands of years. Long ago, games of chance — like rolling dice or drawing lots — were part of everyday life. But nobody had a math system to predict the results.

The study of probability (the math of chance) grew out of real questions about gambling, weather, and fairness. Here are some key moments in its history.

~3000 BCE

Ancient Dice Games

People in Mesopotamia and Egypt played games with knuckle bones and early dice. They noticed that some outcomes happened more often, but they had no formulas to explain why.

1494

Pacioli's Puzzle

Italian mathematician Luca Pacioli posed the 'Problem of Points' — how to fairly split a prize in an unfinished game. This puzzle challenged thinkers for over 150 years.

1654

Pascal & Fermat Write Letters

French mathematicians Blaise Pascal and Pierre de Fermat exchanged letters that solved Pacioli's puzzle. Their work is often called the birth of modern probability.

1814

Laplace's Classic Formula

Pierre-Simon Laplace published the simple formula we still use today: probability equals favorable outcomes divided by total outcomes.

So here's the big question probability answers: If I know all the possible results, how likely is the result I care about? That's exactly what you'll learn in this lesson.

SECTION 2

Core Ideas You Need to Know

Before we do any math, let's nail down a few key vocabulary words. These are the building blocks for everything else in this lesson.

Experiment & Outcome

An experiment is any action with uncertain results (like flipping a coin). An outcome is one possible result (like landing on heads).

Sample Space

The sample space is the complete list of every possible outcome. For a coin: {Heads, Tails}. For a six-sided die: {1, 2, 3, 4, 5, 6}.

Event

An event is one or more outcomes you're interested in. Rolling an even number on a die is an event: {2, 4, 6}.

Probability

Probability is a number from 0 to 1 that measures how likely an event is. 0 means impossible, 1 means certain, and 0.5 means a 50-50 chance.

Complement

The complement of an event is everything that is NOT that event. If your event is 'rain,' the complement is 'no rain.' Together they always add up to 1.

✦ KEY TAKEAWAY

Think of probability like a pizza. The whole pizza is 1 (100%). If you eat 3 out of 8 slices, you ate 3/8 of the pizza. The leftover 5/8 is the complement — the part you didn't eat. The eaten part plus the leftover part always equals the whole pizza (1).

SECTION 3

Seeing Probability in Action

Let's look at a picture that shows how probability works with a standard six-sided die. The diagram below shows every outcome in the sample space and highlights a specific event.

The six die faces make up the sample space. The cyan-highlighted faces (2, 4, 6) are the event. The remaining faces (1, 3, 5) are the complement. Notice that the two probabilities add up to 1.

Look at the bottom of the diagram. The probability of rolling an even number (0.5) plus the probability of NOT rolling an even number (0.5) equals exactly 1. This is always true, no matter what event you pick. That's the power of complements!

SECTION 4

The Formulas You'll Use

Now let's write down the two main formulas. Don't worry — they are short and sweet!

PROBABILITY OF A SIMPLE EVENT

P(A) = Number of favorable outcomes ÷ Total number of outcomes

P(A) means "the probability of event A." Favorable outcomes are the outcomes that match what you want. Total outcomes is the size of the sample space.

COMPLEMENT RULE

P(A') = 1 − P(A)

P(A') (read "P of A complement" or "P of NOT A") is the probability that event A does not happen. Just subtract P(A) from 1.

Why does the complement rule work? Because the event and its complement cover every possible outcome. Nothing is left out, and nothing is counted twice. So their probabilities must add up to 1.

PROBABILITY RANGE

0 ≤ P(A) ≤ 1

A probability can never be less than 0 (impossible) or greater than 1 (certain). You can also express it as a percent between 0% and 100%, or as a fraction.

💡 Quick Tip

If a problem asks for the probability that something does NOT happen, try finding the probability that it DOES happen first, then subtract from 1. This shortcut can save a lot of counting!

SECTION 5

Where Does Your Event Fall on the Scale?

Not all events are equally likely. Some are impossible, some are certain, and most are somewhere in between. The diagram below shows a probability number line with real-life examples placed along it.

This number line places probabilities from 0 (impossible) on the left to 1 (certain) on the right. Notice how rolling a 3 (P = 1/6) and NOT rolling a 3 (P = 5/6) are complements that add to 1.

Probability values and what they tell you

Probability Value	What It Means	Fraction Example
0	Impossible — can never happen	0/6 (rolling a 7 on a standard die)
Close to 0	Very unlikely	1/100 (pulling a specific card)
0.5	Equally likely to happen or not	1/2 (flipping heads)
Close to 1	Very likely	5/6 (not rolling a 3)
1	Certain — must happen	6/6 (rolling some number 1–6)

SECTION 6

Step-by-Step Worked Example

Let's walk through a full problem together. Read each step carefully, and notice how we use both the probability formula and the complement rule.

Marble Bag Problem

Step 1 — Read the ProblemA bag contains 3 red marbles, 5 blue marbles, and 2 green marbles. You reach in and grab one marble without looking. What is the probability of drawing a blue marble? What is the probability of NOT drawing a blue marble?

Step 2 — Find the Total OutcomesCount all the marbles: 3 red + 5 blue + 2 green = 10 marbles total. The sample space has 10 outcomes.

Total outcomes = 10

Step 3 — Count Favorable OutcomesWe want blue marbles. There are 5 blue marbles, so there are 5 favorable outcomes.

Favorable outcomes = 5

Step 4 — Use the Probability FormulaP(blue) = favorable ÷ total = 5 ÷ 10 = 5/10. We can simplify 5/10 to 1/2, which equals 0.5 or 50%.

P(blue) = 1/2 = 0.5 = 50%

Step 5 — Apply the Complement RuleThe complement of 'drawing blue' is 'NOT drawing blue.' Using the complement rule: P(not blue) = 1 − P(blue) = 1 − 1/2 = 1/2. This makes sense because the non-blue marbles (3 red + 2 green = 5) equal 5 out of 10, which is also 1/2.

P(not blue) = 1 − 1/2 = 1/2 = 0.5 = 50%

✅ CHECK YOUR WORK

Always double-check: does your answer fall between 0 and 1? And does P(event) + P(complement) = 1? If yes, you're on the right track!

SECTION 7

Why Complements Are Your Secret Weapon

You might wonder: why bother with complements at all? Can't you just count the outcomes directly? You can — but sometimes the complement is way easier to count. The table below shows when each approach works best.

Choosing the best approach

Approach	When to Use It	Example
Count directly	When the event has only a few favorable outcomes that are easy to list.	"What is the probability of rolling a 6?" Just count: 1 out of 6.
Use the complement	When the event has MANY favorable outcomes or says 'at least one.' Count the simpler opposite first.	"What is the probability of NOT rolling a 6?" P(not 6) = 1 − 1/6 = 5/6. Faster than listing 1, 2, 3, 4, 5.

✦ KEY TAKEAWAY

Imagine you want to know how many songs on your playlist are NOT your favorite song. It's much faster to say 'total songs minus 1' than to count every other song. Complements work the same way — find the easy part, then subtract from 1.

⚠️ Watch Out!

The complement rule only works when the event and its complement cover ALL outcomes with no overlap. For example, 'rolling an even number' and 'rolling an odd number' are complements on a standard die because every roll is one or the other, never both.

SECTION 8

From Simple Events to Bigger Ideas

The skills you've learned here are the foundation for more advanced probability topics. Here's a sneak peek at what's coming next in your math journey.

How this lesson connects to future topics

What You Know Now	What's Coming Next
Probability of one simple event (like rolling a 3)	Compound events — the probability of TWO things happening (like rolling a 3 AND flipping heads)
Complement of one event (P(not A) = 1 − P(A))	Using complements with "at least one" problems, which become very powerful in high school
Equally likely outcomes (fair dice, fair coins)	Events that are NOT equally likely — like weighted dice or real-world data
Listing outcomes by hand	Using tree diagrams, tables, and counting rules to organize larger sample spaces

Everything you do in future probability will build on two things you learned today: the basic probability formula and the complement rule. Master these, and the harder stuff will feel much more manageable!

SECTION 9

Practice Problems

Time to practice! Try each problem on your own before looking at the answer. The problems get harder as you go.

PROBLEM 1 — CONCEPTUAL

In your own words, explain what it means for two events to be complements. Give one example of an event and its complement.

PROBLEM 2 — BASIC CALCULATION

A spinner has 8 equal sections numbered 1 through 8. What is the probability of spinning a 5?

PROBLEM 3 — INTERMEDIATE

A bag holds 4 red chips, 6 white chips, and 2 blue chips. What is the probability of NOT drawing a red chip? Use the complement rule.

PROBLEM 4 — APPLIED

A school cafeteria offers 5 different lunch options each day. Today the options are pizza, salad, tacos, soup, and a sandwich. If a student picks randomly, what is the probability that they do NOT get pizza? Express your answer as a fraction, a decimal, and a percent.

PROBLEM 5 — CRITICAL THINKING

Marcus says the probability of rain tomorrow is 0.3 and the probability of no rain is 0.6. His friend Ava says those numbers can't both be right. Who is correct, and why? What should the correct probability of no rain be?

SUMMARY

Lesson Summary

Probability measures how likely an event is, and you calculate it by dividing the number of favorable outcomes by the total number of outcomes in the sample space. Every probability falls between 0 (impossible) and 1 (certain).

The complement of an event is everything that is NOT the event. The complement rule says P(not A) = 1 − P(A). This shortcut is especially handy when counting the event directly would take a long time. Always check that an event and its complement add to 1 — if they don't, double-check your work!