Basic rules and the addition rule

The language

• An experiment is any process with an uncertain outcome.
• The sample space is the set of all possible outcomes.
• An event is a set of outcomes you are interested in — it may contain one outcome or many.
• $P(A)$ denotes the probability of event $A$.

The rules that hold always

• Every probability lies between 0 and 1:

$$0 \leqslant P(A) \leqslant 1$$

• $P(A) = 0$ means the event cannot happen; $P(A) = 1$ means it is certain. An answer outside this range is always an error — check it every time.
• The probabilities of all outcomes in the sample space sum to 1.
• The complement rule. $A'$ (read "not $A$") is everything in the sample space that is not $A$:

$$P(A') = 1 - P(A)$$

• The complement rule is the most useful shortcut in the standard. Whenever a question says "at least one", the fast route is almost always $1 - P(\text{none})$.

The addition rule

• To find the probability that $A$ or $B$ occurs (meaning $A$, or $B$, or both):

$$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$$

• This is given in the Formulae and Tables Booklet.
• Why subtract the overlap. Adding $P(A)$ and $P(B)$ counts everything in both events twice, so the overlap must be removed once.

• Notation you will meet: $A \cup B$ means "$A$ or $B$" (union), $A \cap B$ means "$A$ and $B$" (intersection).

Reading "or" and "and" correctly

• In probability, "or" is inclusive — $A$ or $B$ includes the case where both happen. Everyday English often means "one or the other but not both", and questions exploit this.
• "and" means both occur together.
• If a question genuinely means exactly one of the two:

$$P(\text{exactly one}) = P(A \text{ or } B) - P(A \text{ and } B)$$

"At least one" — use the complement

• The complement of "at least one" is "none", and "none" is usually a single easy calculation.

$$P(\text{at least one}) = 1 - P(\text{none})$$

• For three independent events each with probability $p$ of occurring:

$$P(\text{at least one}) = 1 - (1-p)^3$$

• Doing this the long way — adding the probabilities of exactly one, exactly two and exactly three — is slow and error-prone, and the specification warns that laborious methods may not show the required thinking.

Checking your answer

• Three checks that catch most errors, and take seconds:
  • Is the answer between 0 and 1?
  • Is it sensible in context? A probability of 0.97 for something described as rare is a signal you have inverted something.
  • Do the parts of a partition sum to 1?