Probability

What does P(A) mean?

The probability that event A occurs.

What does P(A') mean?

The probability that event A does NOT occur. P(A') = 1 − P(A).

What does P(A ∩ B) mean?

The probability that BOTH A and B occur.

What does P(A ∪ B) mean?

The probability that A OR B (or both) occurs.

What does P(A | B) mean?

The conditional probability that A occurs GIVEN that B has occurred.

What are mutually exclusive events?

Events that cannot both occur at the same time, so P(A ∩ B) = 0.

Addition rule for mutually exclusive events

P(A ∪ B) = P(A) + P(B).

General addition rule (any two events)

P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

What are independent events?

Events where one occurring does not change the probability of the other. P(A | B) = P(A) and P(A ∩ B) = P(A) × P(B).

Multiplication rule for independent events

P(A ∩ B) = P(A) × P(B).

General multiplication rule (any two events)

P(A ∩ B) = P(A | B) × P(B) = P(B | A) × P(A).

Conditional probability formula

P(A | B) = P(A ∩ B) / P(B), provided P(B) ≠ 0.

What is a sample space diagram?

A diagram listing every possible outcome of an experiment; useful for calculating probabilities by counting.

What is a tree diagram used for?

Showing successive events, with probabilities on branches; multiply along branches and add between branches.

What is a Venn diagram used for?

Showing how events overlap; useful for visualising A ∩ B, A ∪ B, A' and conditional probabilities.

What is a two-way table used for?

Organising outcomes by two categorical variables; helpful for reading off joint, marginal and conditional probabilities.

P(X = x) notation meaning

The probability that the random variable X takes the specific value x.

What does modelling with probability involve?

Setting up a probability model for a real situation, often making simplifying assumptions (e.g. independence, equal likelihood), and being able to critique those assumptions.

How can probability models be made more realistic?

By relaxing simplifying assumptions — e.g. allowing dependence between events, using empirical probabilities, or modelling unequal likelihoods.