The basics
Probability is a number from 0 (impossible) to 1 (certain) measuring how likely an event is. When all outcomes are equally likely:
P(event) = number of favourable outcomes ÷ number of possible outcomes
The set of all outcomes is the sample space. Rolling two dice has 6 × 6 = 36 outcomes.
The rules
Complement. P(not A) = 1 − P(A). If there is a 35 percent chance of rain, there is a 65 percent chance of no rain. It is often the easiest route: "at least one" is 1 − P(none).
Addition rule (OR). P(A or B) = P(A) + P(B) − P(A and B). If A and B are mutually exclusive (cannot both happen), P(A and B) = 0, so you just add.
Conditional probability. P(A | B) = P(A and B) ÷ P(B): the chance of A given B has happened.
Multiplication rule (AND). P(A and B) = P(A) × P(B | A). If A and B are independent (knowing one tells you nothing about the other), this simplifies to P(A) × P(B).
Independent versus mutually exclusive
These are not the same. Mutually exclusive events with positive probabilities are always dependent: if one happens, the other cannot.
Worked example 1: cards
From a standard 52-card deck, what is P(king or heart)?
- P(king) = 4/52, P(heart) = 13/52.
- The king of hearts is both, so P(king and heart) = 1/52.
- P(king or heart) = 4/52 + 13/52 − 1/52 = 16/52 = 4/13 ≈ 0.308.
Without subtracting the overlap, the king of hearts would be counted twice.
Worked example 2: without replacement
Draw two cards in a row without replacement. What is P(both aces)?
- First card is an ace: 4/52.
- Given that, the second is an ace: 3/51 (one ace and one card are gone).
- P(both) = 4/52 × 3/51 = 12/2652 = 1/221 ≈ 0.0045.
Worked example 3: at least one
Flip a fair coin 3 times. What is P(at least one head)?
- P(no heads) = (1/2)³ = 1/8.
- P(at least one head) = 1 − 1/8 = 7/8.
Using a table
A two-way table makes conditional probabilities easy. If 120 of 200 students are women and 30 of those women play a sport, then P(plays sport | woman) = 30/120 = 0.25, restricting attention to the women only.
Common mistakes
- Adding probabilities of events that overlap without subtracting the overlap.
- Multiplying probabilities of dependent events without adjusting the second one.
- Confusing P(A | B) with P(B | A).