Probability quantifies uncertainty. The good news: most homework problems boil down to a small set of rules and a willingness to count carefully. This guide covers the foundation you need before moving on to distributions, hypothesis testing, or Bayesian inference.

What "probability" means

The probability of an event $A$ is

$P(A) = \frac{\text{favourable outcomes}}{\text{total outcomes}}$

assuming all outcomes are equally likely. $P(A) \in [0, 1]$ :

$0$ = impossible.
$1$ = certain.
$0.5$ = a coin flip.

For non-equally-likely outcomes, you assign weights to each outcome (this is what a probability distribution does).

The three core rules

Addition rule (probability of A or B)

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Subtract the intersection so you don't double-count. If $A$ and $B$ are mutually exclusive (can't both happen), the intersection is zero.

Example: drawing a card from a 52-card deck, $P(\text{King or Heart}) = 4/52 + 13/52 - 1/52 = 16/52 = 4/13$ . (One card is both King and Heart, hence the subtract.)

Multiplication rule (probability of A and B)

$P(A \cap B) = P(A) \cdot P(B \mid A)$

If $A$ and $B$ are independent (one doesn't affect the other), $P(B | A) = P(B)$ , simplifying to $P(A) \cdot P(B)$ .

Example: rolling two dice, $P(\text{both 6}) = 1/6 \cdot 1/6 = 1/36$ . (Rolls are independent.)

Conditional probability

$P(B \mid A) = \frac{P(A \cap B)}{P(A)}$

The probability of $B$ given that $A$ has occurred. Foundation of Bayes' theorem and most of inferential statistics.

Example: a card drawn is a face card. What's the probability it's a King?

$P(\text{King and face card}) = 4/52$ .
$P(\text{face card}) = 12/52$ .
$P(\text{King | face}) = (4/52) / (12/52) = 4/12 = 1/3$ .

Counting: permutations and combinations

For $n$ items choose $r$ :

Permutations (order matters): $P(n, r) = \frac{n!}{(n-r)!}$ .
Combinations (order doesn't): $C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}$ .

The decision is "does swapping two of my chosen items give a different result?":

Yes (e.g. gold vs silver medal) → permutation.
No (e.g. choose a 5-person committee) → combination.

Worked example: lottery

Pick 6 numbers from 49. The order on your ticket doesn't matter — combination.

$\binom{49}{6} = \frac{49!}{6! \cdot 43!} = 13,983,816$

So $P(\text{winning a 6-number jackpot}) = 1/13{,}983{,}816 \approx 7.15 \times 10^{-8}$ .

Independent vs mutually exclusive (don't confuse them!)

Independent: knowing $A$ doesn't change $P(B)$ . Coin flips are independent.
Mutually exclusive: $A$ and $B$ can't both happen. Rolling a die can't be both 1 and 2.

Two events can be one, the other, both, or neither. They are not the same concept, despite being commonly confused.

Common mistakes

The gambler's fallacy: "I've flipped 5 heads in a row, so the next must be tails." Coin flips are independent — the past doesn't change future probability.
Adding non-mutually-exclusive probabilities without subtracting the intersection. $P(\text{King}) + P(\text{Heart}) \neq P(\text{King or Heart})$ .
Conflating $P(A | B)$ with $P(B | A)$ . The classic prosecutor's fallacy: "Given the defendant is innocent, the chance of this evidence is small; therefore given the evidence, the chance of innocence is small." Logically wrong without applying Bayes' theorem.

Try it yourself

Drop any probability problem into the Probability Calculator — addition, multiplication, conditional, with combinatorics. The AI walks you through every step.

Probability Basics: Rules, Combinatorics, and Examples

A clear introduction to probability — definitions, the addition / multiplication / conditional rules, permutations and combinations, and worked examples.