Probability Basics

🎲 Probability Basics

Part 1 of 5 — What Is Probability?

Topics in This Part

🔑 Key Concept: Probability is a number that tells you how likely an event is to happen. It always lands between $0$ (impossible) and $1$ (certain).

What Probability Measures

Some things are sure to happen, some can never happen, and most things are somewhere in between. Probability puts a number on that "in-between."

A few words we'll use the whole lesson:

💡 An event can be one outcome (rolling a $4$) or several outcomes grouped together (rolling an even number: $2$, $4$, or $6$).

The 0-to-1 Scale

Every probability is a number from $0$ to $1$. You can write it as a fraction, a decimal, or a percent.

$0 \le P(\text{event}) \le 1$

The closer the number is to $1$, the more likely the event. The closer to $0$, the less likely.

⚠️ A probability can never be negative and can never be more than $1$. If you ever get an answer like $1.4$ or $-\frac{1}{2}$, you made a mistake somewhere.

Concept Check 🎯

Likely, Unlikely, and Everything Between

We often describe events in words. Here is how the words line up with the scale:

Example: The probability of flipping heads on a fair coin is $\frac{1}{2}$ — an even chance.

Example: The probability that the sun rises tomorrow is essentially $1$ — certain.

Example: The probability of rolling a $7$ on a standard $6$-sided die is $0$ — impossible (there is no $7$).

Match the Description 🔽

Pick the best word for each probability.

Putting Numbers on the Scale

You don't always need a formula to place a probability — sometimes you can reason about how "fair" a situation is.

• A fair coin has two equally likely sides, so $P(\text{heads}) = \frac{1}{2}$.
• A certain event always gets $1$; an impossible one always gets $0$.
• If something is twice as likely to fail as to succeed, success gets the smaller share.

💡 Quick gut-check: before computing, ask "should my answer be near $0$, near $\frac{1}{2}$, or near $1$?" If your final number disagrees with your gut, re-check the count.

Place It on the Scale 🧮

Give each probability as a number from $0$ to $1$.

1) A spinner has $4$ equal sections. $P(\text{landing on one chosen section}) = \,?$ (fraction) 2) $P(\text{a certain event}) = \,?$ 3) $P(\text{an impossible event}) = \,?$

🎲 Probability Basics

Part 2 of 5 — Theoretical Probability & Sample Spaces

🔑 The Formula: When every outcome is equally likely, $P(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}$

The Sample Space

The sample space is the list of all possible outcomes of an experiment. Counting it correctly is the key to every probability problem.

A favorable outcome is any outcome that counts as the event you want.

💡 "Favorable" does not mean "good" — it just means it matches the event you are measuring. If your event is "roll an odd number," the favorable outcomes are $1$, $3$, and $5$.

Theoretical Probability

Theoretical probability is what should happen based on counting, when all outcomes are equally likely.

$P(\text{event}) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$

Worked Example: Rolling an even number on a die

• Total outcomes: $6$ (the numbers $1$ through $6$)
• Favorable outcomes: $3$ (the evens $2$, $4$, $6$)

$P(\text{even}) = \frac{3}{6} = \frac{1}{2}$

Worked Example: Drawing a red marble

A bag has $4$ red, $3$ blue, and $1$ green marble. Total $= 4 + 3 + 1 = 8$.

$P(\text{red}) = \frac{4}{8} = \frac{1}{2}$

🔑 Always simplify your fraction when you can — $\frac{4}{8}$ and $\frac{1}{2}$ mean the same probability.

Concept Check 🎯

Counting Carefully

The whole skill is counting: how many outcomes match the event, and how many outcomes there are in total.

For an event like "greater than $4$" on a die, list them out so you don't miss any: $5$ and $6$ — that's $2$ favorable outcomes out of $6$.

💡 When in doubt, write the sample space ($1,2,3,4,5,6$) and circle the favorable ones. Now try a few on your own.

Find the Probability 🧮

A standard $6$-sided die is rolled. Write each probability as a simplified fraction (like 1/2).

1) $P(\text{rolling a } 5) = \,?$ 2) $P(\text{rolling an odd number}) = \,?$ 3) $P(\text{rolling a number greater than } 4) = \,?$

Spinners Work the Same Way

A spinner with equal sections behaves just like a die: each section is one equally likely outcome.

If a spinner has $8$ equal sections, the total number of outcomes is $8$, and you count favorable sections the same way you counted die faces.

⚠️ This only works when the sections are the same size. Unequal sections are not equally likely, so the simple counting formula would not apply.

Spinner Probabilities 🔽

A spinner has $8$ equal sections numbered $1$ through $8$. Choose the correct probability.

🎲 Probability Basics

Part 3 of 5 — Complements: The Probability of "Not"

🔑 The Complement Rule: The probability that an event does not happen is $P(\text{not } A) = 1 - P(A)$

What Is a Complement?

The complement of an event $A$ is "everything except $A$" — all the outcomes where $A$ does not happen.

Because something either happens or it doesn't, the two probabilities must add to $1$:

$P(A) + P(\text{not } A) = 1$

Rearranging gives the rule:

$P(\text{not } A) = 1 - P(A)$

Worked Example: Not rolling a 6

On a die, $P(\text{rolling a } 6) = \frac{1}{6}$. So:

$P(\text{not } 6) = 1 - \frac{1}{6} = \frac{6}{6} - \frac{1}{6} = \frac{5}{6}$

💡 Why this is handy: counting "all the ways something does NOT happen" can be slow. The complement lets you count the one thing it does, then subtract from $1$.

Worked Example: Marbles

A bag has $10$ marbles: $7$ are red and $3$ are not red.

$P(\text{red}) = \frac{7}{10}$

$P(\text{not red}) = 1 - \frac{7}{10} = \frac{10}{10} - \frac{7}{10} = \frac{3}{10}$

Notice $\frac{7}{10} + \frac{3}{10} = \frac{10}{10} = 1$. ✓

Quick check with percents

If there is a $30\%$ chance of rain, then the chance of no rain is:

$100\% - 30\% = 70\%$

⚠️ The complement of "at least one" is "none," and the complement of "all" is "not all." Read the event carefully before you subtract.

Concept Check 🎯

Two Ways to the Same Answer

For "not red" with $3$ red out of $5$ marbles, you can either:

1. Count directly — there are $2$ non-red marbles, so $\frac{2}{5}$, or
2. Use the complement — $1 - \frac{3}{5} = \frac{2}{5}$.

Both give the same answer. The complement is the faster route when there are many ways to "not" happen. Try it below.

Use the Complement 🧮

Answer with a simplified fraction (like 5/6) unless told otherwise.

1) On a die, $P(\text{not rolling a } 1) = \,?$ 2) A bag is $\frac{3}{8}$ green marbles. $P(\text{not green}) = \,?$ 3) If $P(\text{event}) = 0.4$ as a decimal, then $P(\text{not event}) = \,?$ (decimal)

Quick Recap

The complement rule shows up everywhere — weather, games, marbles, spinners. The setup is always the same:

$P(\text{not } A) = 1 - P(A)$

And the two probabilities always add back to $1$. Use both ideas in the dropdown below.

Complements 🔽

A spinner has $5$ equal sections: $3$ red and $2$ blue. Fill in each probability.

🎲 Probability Basics

Part 4 of 5 — Experimental Probability & Predictions

🔑 Theory vs. Experiment: Theoretical probability comes from counting. Experimental probability comes from actually doing the experiment and recording results.

Experimental Probability

Experimental probability (also called relative frequency) is based on data you collect:

$P(\text{event}) = \frac{\text{number of times the event happened}}{\text{total number of trials}}$

Worked Example: Flipping a coin

Suppose you flip a coin $20$ times and get heads $12$ times.

$P(\text{heads}) = \frac{12}{20} = \frac{3}{5} = 0.6$

The theoretical probability of heads is $\frac{1}{2} = 0.5$. The experiment gave $0.6$ — close, but not exact.

💡 The Law of Large Numbers (in plain words): the more trials you run, the closer your experimental probability usually gets to the theoretical probability.

Worked Example: Reading a results table

A student spun a $4$-color spinner $50$ times:

The total is $15 + 10 + 20 + 5 = 50$. The experimental probability of green is:

$P(\text{green}) = \frac{20}{50} = \frac{2}{5} = 0.4$

And the experimental probability of yellow is:

$P(\text{yellow}) = \frac{5}{50} = \frac{1}{10} = 0.1$

⚠️ For experimental probability, the denominator is the total number of trials ($50$), not the number of colors ($4$).

Concept Check 🎯

Comparing the Two

Use the spinner table from above again:

If the spinner were perfectly fair, each color's theoretical probability would be $\frac{1}{4}$ (one color out of four equal sections). Notice the experimental values are near $\frac{1}{4}$ but not equal — that's normal for only $50$ spins.

💡 The four experimental probabilities should still add to $1$: $\frac{15}{50}+\frac{10}{50}+\frac{20}{50}+\frac{5}{50} = \frac{50}{50} = 1$. ✓

Theoretical vs. Experimental 🔽

A coin is flipped $40$ times and lands heads $24$ times. Fill in each blank.

Making Predictions

You can use probability to predict how many times something will happen in many trials:

$\text{predicted count} = P(\text{event}) \times (\text{number of trials})$

Worked Example

If $P(\text{heads}) = \frac{1}{2}$ and you flip a coin $80$ times, you'd predict:

$\frac{1}{2} \times 80 = 40 \text{ heads}$

Worked Example

If $P(\text{red}) = \frac{1}{4}$ on a spinner and you spin $60$ times, you'd predict:

$\frac{1}{4} \times 60 = 15 \text{ reds}$

💡 A prediction is an estimate. Real results will usually be close but not exactly equal.

Predict the Count 🧮

Use $\text{predicted count} = P \times (\text{trials})$. Enter a whole number.

1) $P(\text{heads}) = \frac{1}{2}$, flip $50$ times. Predict heads: $\,?$ 2) $P(\text{rolling a } 6) = \frac{1}{6}$, roll $60$ times. Predict sixes: $\,?$ 3) $P(\text{blue}) = \frac{1}{4}$, spin $100$ times. Predict blues: $\,?$

🎲 Probability Basics

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) place a probability on the $0$-to-$1$ scale, (2) compute theoretical probability by counting, (3) use the complement rule, and (4) work with experimental probability and predictions. Let's put it all together.

Quick Reference

⚠️ Remember: always simplify fractions, never let a probability exceed $1$ or go below $0$, and an event plus its complement always add to $1$.

Mixed Practice 🎯

One Spinner, Every Skill

The next drill uses a single $10$-section spinner to touch all four skills at once: counting favorable outcomes, simplifying, taking a complement, and predicting a count.

💡 Take them one at a time. For each, decide first: am I counting a probability, subtracting for a complement, or multiplying to predict?

Final Drill 🧮

A spinner has $10$ equal sections numbered $1$ to $10$.

1) $P(\text{landing on a number} > 7)$ as a simplified fraction $= \,?$ 2) $P(\text{NOT landing on } 1)$ as a simplified fraction $= \,?$ 3) If you spin $40$ times, predict the number of times you land on an even number. (whole number) $= \,?$

You're Ready

That's the whole toolkit: the $0$-to-$1$ scale, theoretical probability from counting, the complement rule, and experimental probability with predictions.

The Exit Quiz below pulls one question from each big idea. Read each carefully and decide which tool fits before you answer.

Exit Quiz ✅

Answer all three to finish the lesson.