Probability and Random Variables
What the domain covers
This domain tests sample spaces, events, probability rules, conditional probability, independence, random variables, expectation and variance. The exam emphasises correct set logic and interpretation of a probability distribution.
Core rules
- For equally likely outcomes, P(A) = favourable outcomes/total outcomes.
- Addition: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for mutually exclusive events the last term is zero.
- Multiplication: P(A ∩ B) = P(A)P(B|A); for independent events P(A ∩ B) = P(A)P(B).
- Conditional probability: P(A|B) = P(A ∩ B)/P(B), provided P(B) > 0.
- A random variable assigns a number to each outcome. E(X) = ΣxP(x) for discrete X; Var(X) = E(X²) − [E(X)]².
Worked example
A fair die is rolled. Let A = 'even' = {2,4,6} and B = 'greater than 3' = {4,5,6}. P(A) = 3/6, P(B) = 3/6, A ∩ B = {4,6}, so P(A ∪ B) = 3/6 + 3/6 − 2/6 = 4/6 = 2/3. For a random variable with values 0,1,2 and probabilities 0.2,0.5,0.3, E(X) = 0(0.2) + 1(0.5) + 2(0.3) = 1.1.
Traps
Mutually exclusive events cannot occur together; independent events can. Do not add probabilities without subtracting the overlap. In conditional probability, divide by P(B), not P(A). Check that probabilities sum to 1 and that Var(X) is non-negative. 'At least one' is often easier through 1 − P(none).