Summary — What This Chapter Covers
- A probability distribution table lists each value of X with its probability; ΣP(X = x) = 1.
- E(X) = Σx p, Var(X) = Σx2p − [E(X)]2.
- Binomial B(n, p): fixed number of independent trials, two outcomes, constant p.
- Geometric Geo(p): number of trials up to and including the first success; P(X = r) = p(1 − p)r−1.
- B(n, p): E = np, Var = np(1 − p). Geo(p): E = 1/p.
1. Probability Distributions
Distribution tables
Building the table
Identify every possible value of X, find each probability (often with a tree or by counting), and check that they sum to 1. If the probabilities are given by a formula with an unknown constant, use Σp = 1 to find it.
Expectation and variance
E(X) = Σxp Var(X) = Σx2p − [E(X)]2
Worked Example
A bag contains 3 red and 2 green counters. Two are taken without replacement; X is the number of red counters. Find the distribution of X, E(X) and Var(X).
Practice Question 1
P(X = x) = kx for x = 1, 2, 3, 4. Find k, E(X) and Var(X).
2. Binomial and Geometric Distributions
B(n, p) and Geo(p)
Recognising the model
Binomial X ~ B(n, p) counts successes in a fixed number n of independent trials, each with the same probability p of success. Geometric X ~ Geo(p) counts the number of trials up to and including the first success, with independent trials and constant p. State the conditions in context when asked whether a model is suitable.
For geometric "at most" and "more than" probabilities, P(X > r) = (1 − p)r (the first r trials all fail), so P(X ≤ r) = 1 − (1 − p)r.
Formulae
B(n, p): P(X = r) = nCr pr(1 − p)n−r Geo(p): P(X = r) = p(1 − p)r−1, r = 1, 2, 3, …
Worked Example 1
X ~ B(10, 0.3). Find P(X = 3) and P(X ≤ 1).
Worked Example 2
A fair die is thrown until a six appears. Find the probability that the first six is on the 4th throw, and the probability that it takes more than 5 throws.
3. Expectation and Variance
Mean and variance
Shortcut formulae
Memorise
Table
Probabilities sum to 1.
Binomial
Fixed n, independent, constant p. Mean np, var np(1 − p).