Learn: Probability

A quick, calm intro — then you choose when to climb.

The idea

Probability measures chance — how likely a result is, from 0 (never) to 1 (always). It runs on one fraction and two moves.

The two moves:

1
P = good ÷ all — every chance starts as a fraction — count the outcomes you want, divide by every possible outcome
2
AND → multiply — this AND that both happen — independent events multiply their chances
3
OR → add — either this or that happens — add the chances when the events do not overlap (they cannot both happen)

Quick memory tips

A percent is just the fraction × 100 — 1/4 = 25%
“Both” is rarer than either alone: a product of fractions shrinks
S = sample space (all outcomes) · A = event (the ones you want) · P(A) = n(A) ÷ n(S)

Set notation

The three symbols that carry every probability problem.

$S$

sample space — every possible outcome (the 12 tickets)

$A$

event — the outcomes we want (the 3 winners)

$P(A) = \frac{n(A)}{n(S)}$

good ÷ all — lucky dip: 3 ÷ 12 = 1/4

n(S) = 12 · n(A) = 3 → $P(A) = \frac{3}{12} = \frac{1}{4}$

Worked example

Every step explained — why we do it, and what happens.

1
$P = \frac{3}{12} = \frac{1}{4}$
Why: Single event: 3 winning tickets out of 12 — the good outcomes over every outcome: n(A) over n(S).
What happens: P = 1/4 — the fraction of chance.
2
$= 25\%$
Why: A percent is the fraction × 100: 1/4 = 0.25 → 25%.
What happens: 25% — the same chance written “out of 100”.
3
$\frac{1}{2} \times \frac{1}{2} = \frac{1}{4} = 25\%$
Why: Independent AND: two coins both landing heads — two independent events, and “both” is AND, so multiply the separate chances.
What happens: 25% — rarer than either coin alone, because both must cooperate.
4
$\text{Trap: } \frac{1}{2} + \frac{1}{2} = 1 = 100\%\,?$
Why: The trap: adding the chances. Adding is for OR — this OR that, when the events do not overlap. “Both” is AND, and AND multiplies.
What happens: 100% would be a guarantee — two coin flips can never promise double heads.
5
$\text{AND} \to \text{multiply} \qquad \text{OR} \to \text{add}$
Rule summary: AND → multiply (independent events) · OR → add (events that do not overlap).

Counting (Optional Toolbox)

Fast ways to count without listing every outcome. Not needed for the basics — come back when the questions get big.

optional

Factorial

n! = n \times (n-1) \times \cdots \times 1

Multiply the countdown: $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$ — five books on a shelf, spot by spot.

Permutations (order matters)

P(n, r) = \frac{n!}{(n - r)!}

Arrangements: who is first changes the answer. Five books on a shelf → $5! = 120$ orderings.

Combinations (order does not)

C(n, r) = \frac{n!}{r!(n - r)!}

Selections: the team is the team whoever you pick first. A team of 3 from 5 → $C(5, 3) = 10$ .

Shelf = order matters → permutation · team = order does not → combination. The factorial is the building block of both.

Real-world example

The school fair has a lucky dip: a jar holds 12 tickets and 3 of them win prizes. A second jar holds 5 tickets with 1 winner. What is the chance of grabbing a winner from the first jar — and of winning BOTH jars?

$\text{both jars: } \frac{3}{12} \times \frac{1}{5} = \frac{1}{20} = 5\%$

Follow the win → win path: both jars is one path, and one path multiplies.

1
$P_1 = \frac{3}{12}$
Why: 3 winning tickets out of 12 in the jar — n(A) over n(S): good outcomes over all outcomes.
What happens: P₁ = 3/12.
2
$= \frac{1}{4} = 25\%$
Why: Simplify 3/12 to 1/4; × 100 gives the percentage.
What happens: 25% on the first jar.
3
$\text{Trap: } \frac{1}{4} + \frac{1}{5} = \frac{9}{20} = 45\%\,?$
Why: The trap: adding the chances. The tree shows both jars as ONE path — win, then win again — and one path multiplies.
What happens: 45% would be “first OR second” — not “both”.
4
$\frac{1}{4} \times \frac{1}{5} = \frac{1}{20} = 5\%$
Why: Both jars: multiply. The second jar has 1 winner in 5, so 1/4 × 1/5.
What happens: 5% — one chance in twenty for the double win.
5
$\frac{1}{20} < \frac{1}{4} \text{ and } \frac{1}{5}$
Why: The check: two wins must be rarer than either win alone — a product of fractions shrinks, a sum would grow.
What happens: 5% < 25% and 5% < 20% ✓ — multiplication was right.

Common glitch

Two classic glitches:

1.AND vs OR — adding independent chances

Wrong:P(both) = P(A) + P(B) — two chances, so add them

Right:“Both” is AND → multiply. Addition is for OR — either one — and only when the events do not overlap.

2.Turning a fraction into a percent

Wrong:3/12 = 3% — just swap the fraction for a percent

Right:Percent = fraction × 100: 3/12 = 1/4 = 25%. “Per cent” means “out of 100”.

Probability Calculator — Count outcomes and calculate simple probabilities.
Percentage Calculator — Express a probability as a percentage.

Next up: Statistics →

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Common glitch

Two classic glitches:

1.AND vs OR — adding independent chances

Wrong:P(both) = P(A) + P(B) — two chances, so add them

Right:“Both” is AND → multiply. Addition is for OR — either one — and only when the events do not overlap.

2.Turning a fraction into a percent

Wrong:3/12 = 3% — just swap the fraction for a percent

Right:Percent = fraction × 100: 3/12 = 1/4 = 25%. “Per cent” means “out of 100”.

Learn: Probability

The idea

The two moves:

Quick memory tips

Worked example

Real-world example

Common glitch

Try a related calculator

Learn: Probability

The idea

The two moves:

Quick memory tips

Worked example

Real-world example

Common glitch

Try a related calculator