Ch 4 · 5.4 Discrete Random Variables · A Level Mathematics 9709
Topic 4 of 5 in Probability & Statistics 1 — A Level Mathematics 9709 — 61 lessons.
Introduction
How many of 10 seeds will germinate? How many penalty kicks until the first goal? A random variable turns the outcome of a random process into a number, and its probability distribution lists the possible values with their probabilities. Two distributions appear so often that they have names: the binomial and the geometric. (Syllabus section 5.4, Paper 5.)
Key ideas — Probability distributions
A discrete random variable X takes separate values x_1, x_2, dots with probabilities p_1, p_2, dots that add to 1. Write the distribution as a table. To build one, work out the probability of each value (often with a tree diagram), and check the total.
🧑🏫 Teacher's explanation
Imagine you are playing a board game where you roll a fair six-sided die. The number you roll is a 'discrete random variable' because you can only land on specific, separate integers like 1, 2, or 3—you can't roll a 2.5. Each outcome has a specific probability attached to it, and if you add up the probabilities of every possible outcome, they must equal exactly 1 because something has to happen.
📖 Definitions
🔍 Worked example — Easy
A discrete random variable X has the following probability distribution: \begin{array}{|c|c|c|c|c|}\hline x & 1 & 2 & 3 & 4 \\ \hline P(X=x) & 0.1 & 0.3 & 0.4 & 0.2 \\ \hline \end{array}$ Calculate $E(X)$.
🔍 Worked example — Standard
A fair coin is tossed 3 times. Let X be the number of heads obtained. Find E(X) and Var(X) using the binomial distribution B(n, p) formulas.
🔍 Worked example — Exam-style
A machine produces faulty items with a probability of 0.05. Let X be the number of trials until the first faulty item is produced. Find E(X) and Var(X) using the geometric distribution Geo(p).
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Expectation and variance
E(X) = Σ x p Var(X) = Σ x^2 p - big[E(X)big]^2
🧑🏫 Teacher's explanation
Imagine you are playing a carnival game where you win different amounts of money depending on which prize you land on. If you played this game thousands of times, the average amount you win per game is the 'Expectation'. It is not necessarily a value you can actually win in a single game (like winning $1.50 when prizes are only $1 or $2), but it is the long-term average.
📖 Definitions
🔍 Worked example — Easy
A discrete random variable X has the following probability distribution: x: 1, 2, 3 P(X=x): 0.2, 0.5, 0.3 Calculate E(X) and Var(X).
🔍 Worked example — Standard
A discrete random variable X has the following distribution: x: 0, 1, 2, 3 P(X=x): 0.1, k, 0.3, 0.2 Find the value of k and the expectation E(X).
🔍 Worked example — Exam-style
A discrete random variable X has E(X) = 4 and E(X^2) = 20. Calculate the standard deviation of X.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
x: 0, 1, 2, 3, 4 P(X=x): 0.1, 0.2, 0.4, 0.2, 0.1 Calculate the variance of X. [4 marks]
Key ideas — The binomial distribution
X sim B(n, p) counts the number of successes in n trials when
🧑🏫 Teacher's explanation
Imagine you are playing a game where you flip a coin 10 times and want to know the chance of getting exactly 3 heads. The Binomial Distribution is the mathematical tool that tells you exactly how likely that specific outcome is. It doesn't just look at one event, but the combination of many identical events.
📖 Definitions
🔍 Worked example — Standard
A machine produces components where 3% are defective. If 50 components are checked, find the expected number of defective components and the variance.
🔍 Worked example — Exam-style
In a large population, 10% of people are left-handed. A sample of 12 people is taken. Find the probability that at least 1 person is left-handed.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — The geometric distribution
X sim Geo(p) counts the number of trials up to and including the first success, under the same conditions but with no fixed number of trials:
🧑🏫 Teacher's explanation
Imagine you are playing a game of 'heads or tails' with a coin, and you keep flipping until you finally see a 'heads'. You don't know if it will take one flip, five flips, or fifty flips. The Geometric distribution models exactly this: the number of trials needed to get your very first success. Unlike the Binomial distribution, where you have a fixed number of attempts (like 10 coin flips), the Geometric distribution has no upper limit; you just keep going until you succeed.
📖 Definitions
🔍 Worked example — Easy
A marksman has a probability 0.8 of hitting a target. Let X be the number of shots taken until he hits the target for the first time. Find P(X = 3).
🔍 Worked example — Standard
A machine produces faulty parts with probability 0.05. Let X be the number of parts produced until the first faulty part is found. Find P(X > 4).
🔍 Worked example — Exam-style
In a large population, 10\% of people are left-handed. Let X be the number of people sampled until the first left-handed person is found. Find the expected number of people sampled.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Worked examples — Example 1: building a distribution
Two balls are taken without replacement from a bag of 3 red and 2 blue. X is the number of red balls taken.
🔍 Worked example — Easy
Two balls are taken without replacement from a bag containing 3 red and 2 blue balls. Let X be the number of red balls taken. Construct the probability distribution table and find E(X).
🔍 Worked example — Standard
Using the distribution from the previous example, calculate Var(X).
🔍 Worked example — Exam-style
A discrete random variable Y has the following probability distribution: P(Y=y) = \0.1, 0.3, 0.4, 0.2\ for y = \1, 2, 3, 4\. Find E(Y) and Var(Y).
⚠️ Common errors
Worked examples — Example 2: binomial
30% of a town's residents cycle to work. In a random sample of 10, find the probability that exactly 3 cycle, and that at most 2 cycle. X sim B(10, 0.3):
🔍 Worked example — Easy
A fair six-sided die is rolled 5 times. Let X be the number of times a '6' is rolled. Find P(X=2) and E(X).
🔍 Worked example — Standard
In a sample of 10 residents, 30% cycle to work. Let X be the number of cyclists. Calculate P(X ≤ 2) and Var(X).
🔍 Worked example — Exam-style
A machine produces components with a 5% defect rate. In a batch of 20 components, find the probability that exactly 3 are defective and find the expected number of defective components.
⚠️ Common errors
Worked examples — Example 3: geometric
A fair die is rolled until a six appears. X sim Geo (tfrac16 ). P(X = 4) = (tfrac56 )^3tfrac16 = 0.0965; P(X > 3) = (tfrac56 )^3 = 0.579; E(X) = 6 rolls.
🔍 Worked example — Easy
A fair six-sided die is rolled until a 6 appears. Let X be the number of rolls required. Find P(X=4).
🔍 Worked example — Exam-style
In a factory, the probability that a component is defective is 0.05. Let X be the number of components tested until the first defective one is found. Find E(X).
⚠️ Common errors
Worked examples — Example 4: an unknown in a table
X takes values 1, 2, 3 with probabilities 0.2, a, b, and E(X) = 2.3. 0.2 + a + b = 1 and 0.2 + 2a + 3b = 2.3. So a + b = 0.8 and 2a + 3b = 2.1, giving b = 0.5, a = 0.3.
🔍 Worked example — Easy
X takes values 1, 2, and 3 with probabilities 0.2, a, and b respectively. Given E(X) = 2.3, find a and b.
🔍 Worked example — Standard
X can take values 0, 1, 2, 3. The probabilities are 0.1, 0.3, p, q. Given E(X) = 1.5, find p and q.
🔍 Worked example — Exam-style
A discrete random variable X has the following probability distribution: \n| x | 1 | 2 | 3 | 4 |\n|---|---|---|---|---|\n| P(X=x) | 0.1 | k | 2k | 0.2 |\nFind the value of k and calculate E(X).
⚠️ Common errors
Common mistakes
Practice
Answers: (1) binom82(0.25)^2(0.75)^6 = 0.311; 1 - 0.75^8 = 0.900. (2) n(0.4)(0.6) = 3.84 ⇒ n = 16; E(X) = 6.4. (3) 0.3^2 × 0.7 = 0.063; E(X) = 1/0.7 = 1.43. (4) 1 - (tfrac56 )^n > 0.9 ⇒ (tfrac56 )^n < 0.1; n = 13 ((tfrac56 )^12 = 0.112, (tfrac56 )^13 = 0.093). (5) E(X) = tfrac12 + tfrac23 + tfrac23 = 11/6; Σ x^2p = tfrac12 + tfrac43 + 16/6 = 4.5; Var(X) = 4.5 - 121/36 = 41/36 = 1.14.