Skill: probability-basics

Probability Basics

Probability is a number between 0 and 1 that measures how likely something is — favorable outcomes over total outcomes. Flip coins, roll dice, and spin spinners with confidence.

1 Understand

One formula — what you want over everything possible — answers every theoretical probability question.

What it is

Probability measures how likely an event is, as a number between 0 (impossible) and 1 (certain): P(event) = favorable outcomes ÷ total possible outcomes. It works as a fraction, a decimal, or a percent — 1/2 = 0.5 = 50% all say the same thing.

Why it matters

Probability is the language of uncertainty: weather forecasts, game odds, medical risks, and polling margins all speak it. Every statistics idea after this — sampling, testing, prediction — builds on P(event).

Where it is used

Games: dice, coins, cards, spinners
Weather: “40% chance of rain”
Medicine: risk of side effects
Polls and election forecasts

2 See It

Two pictures that do the counting for you.

3 red sections out of 8 equal sections — P(red) = 3/8. Equal sections mean equally likely outcomes.

Count what you want over everything

The spinner has 8 equal sections, so every section is equally likely. 3 of them are red, so P(red) = 3/8. The “equal” part matters: if sections had different sizes, you could not just count them.

Tree diagrams list every path

For two coins, draw the tree: first flip splits into H and T, each splits again. Four equal paths: HH, HT, TH, TT — each 1/2 × 1/2 = 1/4. “One heads and one tails” covers two paths (HT and TH), so its probability is 2/4 = 1/2.

The tree for two coins: four equally likely paths, each worth 1/4. Multiply along a path; add across the paths you want.

3 Worked Examples

Follow each step. The pattern is always the same: count favorable, count total, divide, simplify.

Example 1 Die: P(rolling an even number)

Favorable faces: 2, 4, 6 — that is 3 out of 6.
P(even) = 3/6 = 1/2.

P = 1/2 = 50%

Simplify the fraction — 3/6 means the same thing, but 1/2 is the standard answer.

Example 2 Complement: P(not rolling a 6)

P(6) = 1/6, and everything must sum to 1.
P(not 6) = 1 − P(6) = 1 − 1/6 = 5/6.

P = 5/6

Direct count agrees: five non-6 faces out of six.

Example 3 Independent events: ace, replace, then king

P(ace) = 4/52 = 1/13. Replacing resets the deck, so P(king) = 1/13 too.
Independent events multiply: P(both) = 1/13 × 1/13 = 1/169.

P = 1/169

Trap check: adding gives 2/13 — but “AND” (both happen) multiplies, “OR” adds.

4 Common Mistakes

Three errors that show up on almost every probability quiz.

Mistake 1: Forgetting the outcomes must be equally likely

Wrong

“A bag has 1 red and 9 blue marbles, so P(red) = 1/2 — red or blue.”

Right

Count individual equally likely outcomes: P(red) = 1/10.

Rule: favorable over total only works when every outcome is equally likely.

Mistake 2: Adding when you should multiply (and vice versa)

Wrong

P(ace then king) = 1/13 + 1/13 = 2/13.

Right

“Both happen” = AND = multiply: 1/13 × 1/13 = 1/169. “One or the other” = OR = add.

Rule: AND multiplies, OR adds. Check which word the question is asking.

Mistake 3: Leaving the fraction unsimplified

Wrong

P(face card) = 12/52 as the final answer.

Right

12/52 = 3/13 — always reduce to simplest form.

Rule: simplify the probability, just like any fraction.

5 Quick Check

Try each one on paper first, then reveal the answer.

1. A fair coin is flipped. What is P(heads)?

Answer

One favorable outcome out of two: P(heads) = 1/2 = 50%.

2. Two fair coins are flipped. What is P(both heads)?

Answer

Four paths (HH, HT, TH, TT); only HH works: P = 1/4.
Equivalently, 1/2 × 1/2 = 1/4.

3. A bag has 2 red and 3 blue marbles. What is P(not red)?

Answer

P(red) = 2/5, so P(not red) = 1 − 2/5 = 3/5.
Direct count agrees: 3 blue marbles out of 5.

Key Points to Remember

P(event) = favorable outcomes ÷ total possible outcomes.
The formula only works when every outcome is equally likely.
Complement: P(not E) = 1 − P(E).
AND multiplies (independent events); OR adds (mutually exclusive events).
Simplify the probability, just like any fraction.

Practice probability Try the probability tool See probability on FiboMap