1 Reading the notation

Probability notation is compact and easy to misread. Each symbol below changes the meaning of a statement completely.

8 terms

Cover the right column and check you can say what each one means.

P(A): “The probability that event A happens.” Always between 0 and 1.
P(A and B) (∩, intersection): Both A and B happen. Multiply, but only directly if the events are independent.
P(A or B) (∪, union): At least one happens. P(A) + P(B) − P(A and B). The subtraction stops the overlap being counted twice.
P(A | B) (the vertical bar): “The probability of A GIVEN THAT B has happened.” The bar restricts you to the cases where B is true. P(A|B) and P(B|A) are different numbers with different denominators. Swapping them is the base-rate fallacy.
P(A′) or P(not A): The complement. A does not happen. P(not A) = 1 − P(A).
Mutually exclusive: Cannot both happen: P(A and B) = 0. Not the same as independent, in fact mutually exclusive events with non-zero probability are always DEPENDENT.
Independent: One happening does not change the probability of the other: P(A and B) = P(A) × P(B).
Relative risk: The RATIO of risk in one group to another. Says nothing about the ABSOLUTE size of either risk. A doubled relative risk on a 1-in-a-million baseline is still nearly nothing.

✓

P(A|B) reads left-to-right as “A given B”. The thing AFTER the bar is what you already know; the thing BEFORE it is what you are asking about.

Practise just this