Probability for Students: Three Rules That Cover Most Questions

Probability has more bad notation than any other school topic. The actual ideas are simple. Three rules cover most exam questions.

1. The complement rule

P(not A) = 1 - P(A)

If the chance of rain is 0.3, the chance of no rain is 0.7. That obvious fact is one of your most useful tools — when computing "at least one" type questions, find the chance of zero and subtract from 1.

2. The addition rule

P(A or B) = P(A) + P(B) - P(A and B)

The minus term catches the overlap. If you forget it, you double-count people who fit both A and B.

For mutually exclusive events (can't happen together) the overlap is 0, so it simplifies to P(A) + P(B).

3. The multiplication rule

P(A and B) = P(A) × P(B|A)

P(B|A) is "probability of B given A already happened". For independent events (one doesn't change the other), P(B|A) = P(B).

Where students lose marks

• Treating events as independent when they aren't (drawing cards without replacement is dependent)
• Forgetting the overlap in the addition rule
• Confusing "at least one" with "exactly one"

A useful trick

For "at least one" questions:

P(at least one) = 1 - P(none)

It's almost always easier to compute the chance of zero than the chance of one or more.

Tree diagrams

For multi-step probability, draw a tree. Each branch shows a probability. Multiply along a branch, add between branches. This visual catches mistakes that pure algebra hides.

Practice probability questions →