Probability and Probability Distributions
Probability questions combine counting, conditional probability and the standard models - binomial, Poisson and normal - including the Poisson approximation to the binomial. The examiner expects the model to be named and its parameters justified before any calculation begins.
Basic rules
- P(A ∪ B) = P(A) + P(B) − P(A ∩ B); mutually exclusive events have zero intersection.
- P(A|B) = P(A ∩ B)/P(B); independence means P(A ∩ B) = P(A)P(B).
- On a tree diagram, multiply along branches and add across mutually exclusive outcomes.
- Counting: arrangements n!/(n₁!n₂!…), selections C(n, r), ordered choices P(n, r).
Discrete random variables
- E(X) = Σx P(X = x) and Var(X) = E(X²) − [E(X)]².
- E(aX + b) = aE(X) + b, while Var(aX + b) = a²Var(X).
Standard distributions
- Binomial B(n, p): fixed n, independent trials, constant p; P(X = r) = C(n, r)p^r(1 − p)^(n−r), with mean np and variance np(1 − p).
- Poisson Po(λ): events in a fixed interval at a constant average rate; mean = variance = λ and P(X = r) = e^(−λ)λ^r/r!. Use it as an approximation to the binomial when n is large and p is small with np moderate.
- Normal N(μ, σ²): standardise with z = (x − μ)/σ, read the lower tail from tables and use symmetry for the upper tail.
Worked example
X ~ B(10, 0.3). Find P(X ≥ 2). Using the complement, P(X ≥ 2) = 1 − P(0) − P(1) = 1 − 0.7¹⁰ − 10(0.3)(0.7⁹) = 1 − 0.02825 − 0.12106 = 0.8507, to four decimal places.
Traps
- Using the binomial when trials are not independent, or the Poisson where the variance clearly differs from the mean.
- Omitting the continuity correction when a normal approximates a discrete distribution.
- Reporting a probability outside [0, 1], which signals that a complement was mishandled.