Probability Mind Map

For two events A and B with P(B) > 0, the conditional probability of A given B is P(A|B) = P(A∩B)/P(B). It measures the chance of A after restricting the sample space to B.

Before substituting, underline the phrase after 'given that'. That event must go in the denominator of the conditional probability formula.

For two events A and B, P(A∩B) = P(A)P(B|A) when P(A) > 0, and also P(A∩B) = P(B)P(A|B) when P(B) > 0.

Look for words such as 'both', 'and', 'successively', or 'without replacement'. These often require the multiplication theorem.

Two events A and B are independent if the occurrence of one does not change the probability of the other. Equivalently, P(A∩B)=P(A)P(B), provided the relevant probabilities are defined.

To test independence, compare P(A∩B) with P(A)P(B). Do not decide from wording alone.

If E1, E2, ..., En are mutually exclusive and exhaustive events with P(Ei)>0, then for any event A, P(A)=Σ P(Ei)P(A|Ei).

First check that the cases form a partition: no overlap and together cover the whole sample space.

If E1, E2, ..., En form a partition of the sample space and A is an event with P(A)>0, then P(Ei|A)=P(Ei)P(A|Ei)/ΣP(Ej)P(A|Ej).

Name the observed event as A and the possible sources as E1, E2, ..., En. The required source after observation goes in the numerator.

A random variable is a real-valued function on the outcomes of a random experiment. For a discrete random variable X, its probability distribution lists values xi with probabilities pi such that pi≥0 and Σpi=1.

Make a table with columns xi, pi, xipi, and xi^2pi. This prevents missing terms in E(X) and Var(X).