What you'll learn
This revision guide covers probability distributions as specified in the AQA GCSE Statistics syllabus. You'll learn to work with discrete probability distributions, calculate expected values and measures of spread, and understand when to apply the binomial distribution. These skills are essential for Paper 2 questions worth 6-12 marks.
Key terms and definitions
Probability distribution — a table, function or graph showing all possible outcomes of a random variable and their associated probabilities
Random variable — a variable whose value depends on the outcome of a random event, typically denoted by X
Expected value (mean) — the long-run average value of a random variable, calculated as E(X) = Σxp(x), where x represents each outcome and p(x) its probability
Discrete probability distribution — a probability distribution where the random variable can only take specific, countable values (like 0, 1, 2, 3...)
Binomial distribution — a discrete probability distribution for the number of successes in a fixed number of independent trials, each with the same probability of success
Variance — a measure of spread for a probability distribution, calculated as Var(X) = E(X²) - [E(X)]²
Standard deviation — the square root of the variance, measuring the typical distance of values from the expected value
Trial — a single performance of an experiment with two or more possible outcomes
Core concepts
Understanding discrete probability distributions
A discrete probability distribution must satisfy two conditions:
- All probabilities must be between 0 and 1 inclusive: 0 ≤ p(x) ≤ 1
- The sum of all probabilities must equal 1: Σp(x) = 1
When presented with a probability distribution table, always check these conditions first. If probabilities contain an unknown value (like k or a), use the condition that probabilities sum to 1 to find it.
Example structure of a discrete probability distribution:
| X | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| P(X = x) | 0.1 | 0.3 | 0.4 | 0.2 |
This shows that the random variable X can take values 0, 1, 2, or 3, with their respective probabilities. Notice that 0.1 + 0.3 + 0.4 + 0.2 = 1.
Calculating expected value (mean)
The expected value E(X) represents the theoretical mean of the distribution. It's what you'd expect the average to be after many repetitions.
Formula: E(X) = Σxp(x)
Method:
- Multiply each outcome value by its probability
- Add all these products together
For the distribution above: E(X) = (0 × 0.1) + (1 × 0.3) + (2 × 0.4) + (3 × 0.2) = 0 + 0.3 + 0.8 + 0.6 = 1.7
The expected value doesn't have to be a possible outcome. Here, you can't actually score 1.7, but it represents the long-run average.
Calculating variance and standard deviation
Variance measures how spread out the distribution is from its mean. There are two equivalent formulas:
Formula 1: Var(X) = Σx²p(x) - [E(X)]²
Formula 2: Var(X) = E(X²) - μ² where μ = E(X)
Method for Formula 1:
- Calculate E(X²) by multiplying each x² value by its probability p(x) and summing
- Calculate [E(X)]² by squaring the expected value
- Subtract: Var(X) = E(X²) - [E(X)]²
The standard deviation is simply:
σ = √Var(X)
Use standard deviation when you need the measure of spread in the same units as the original data.
The binomial distribution
The binomial distribution models situations where:
- There are a fixed number of trials (n)
- Each trial has exactly two outcomes (success or failure)
- Each trial is independent
- The probability of success (p) is constant for each trial
We write: X ~ B(n, p)
This reads as "X follows a binomial distribution with n trials and probability p"
Key binomial formulas you must know:
For X ~ B(n, p):
- E(X) = np
- Var(X) = np(1 - p) or npq where q = 1 - p
- Standard deviation = √[np(1-p)]
When to use binomial:
- Counting the number of successes in repeated trials
- Fixed number of attempts
- Independent events
- Constant probability
When NOT to use binomial:
- Sampling without replacement from small populations (probabilities change)
- More than two possible outcomes per trial
- Variable number of trials
Calculating binomial probabilities
For individual probabilities, use the formula:
P(X = r) = ⁿCᵣ × pʳ × (1-p)ⁿ⁻ʳ
Where:
- ⁿCᵣ means "n choose r" = n!/(r!(n-r)!)
- r is the number of successes
- n is the number of trials
- p is the probability of success
At GCSE level, you'll typically calculate:
- P(X = r) for specific values
- P(X ≤ r) or P(X ≥ r) by adding individual probabilities
- Use of probability tables provided in the exam
Most AQA papers provide binomial probability tables, so you'll read values rather than calculate from the formula directly.
Interpreting binomial probability tables
AQA provides cumulative probability tables showing P(X ≤ x).
Key reading skills:
To find P(X = 5) when X ~ B(10, 0.3):
- Read P(X ≤ 5) from the table
- Read P(X ≤ 4) from the table
- Calculate: P(X = 5) = P(X ≤ 5) - P(X ≤ 4)
To find P(X > 6):
- Read P(X ≤ 6) from the table
- Calculate: P(X > 6) = 1 - P(X ≤ 6)
To find P(X ≥ 7):
- Read P(X ≤ 6) from the table
- Calculate: P(X ≥ 7) = 1 - P(X ≤ 6)
Always check carefully whether the question asks for >, ≥, <, or ≤.
Worked examples
Example 1: Finding unknown probabilities (4 marks)
Question: The discrete random variable X has the probability distribution shown below:
| X | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| P(X = x) | 0.2 | k | 2k | 0.3 |
(a) Find the value of k. (2 marks) (b) Calculate P(X ≥ 3). (2 marks)
Solution:
(a) Sum of all probabilities must equal 1: 0.2 + k + 2k + 0.3 = 1 0.5 + 3k = 1 ✓ 3k = 0.5 k = 0.5 ÷ 3 = 1/6 or 0.167 (3 s.f.) ✓
(b) P(X ≥ 3) = P(X = 3) + P(X = 4) = 2k + 0.3 ✓ = 2(1/6) + 0.3 = 1/3 + 0.3 = 0.633 (3 s.f.) or 19/30 ✓
Example 2: Expected value and variance (6 marks)
Question: A game involves rolling a biased die. The score X and probabilities are:
| X | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| P(X = x) | 0.15 | 0.20 | 0.25 | 0.20 | 0.12 | 0.08 |
(a) Calculate E(X). (3 marks) (b) Calculate Var(X). (3 marks)
Solution:
(a) E(X) = Σxp(x) = (1 × 0.15) + (2 × 0.20) + (3 × 0.25) + (4 × 0.20) + (5 × 0.12) + (6 × 0.08) ✓ = 0.15 + 0.40 + 0.75 + 0.80 + 0.60 + 0.48 ✓ = 3.18 ✓
(b) First calculate E(X²): E(X²) = (1² × 0.15) + (2² × 0.20) + (3² × 0.25) + (4² × 0.20) + (5² × 0.12) + (6² × 0.08) ✓ = 0.15 + 0.80 + 2.25 + 3.20 + 3.00 + 2.88 = 12.28
Var(X) = E(X²) - [E(X)]² ✓
= 12.28 - (3.18)²
= 12.28 - 10.1124
= 2.17 (3 s.f.) ✓
Example 3: Binomial distribution application (6 marks)
Question: In a certain school, 35% of students walk to school. A random sample of 12 students is selected.
(a) Give two reasons why a binomial distribution might be a suitable model. (2 marks) (b) Calculate the expected number of students who walk to school. (2 marks) (c) Calculate the standard deviation of the number of students who walk. (2 marks)
Solution:
(a) • Fixed number of trials (n = 12 students) ✓ • Probability of walking is constant for each student (p = 0.35) ✓
[Also acceptable: Each student is independent / Only two outcomes per student]
(b) Let X = number of students who walk X ~ B(12, 0.35) E(X) = np ✓ = 12 × 0.35 = 4.2 students ✓
(c) Var(X) = np(1 - p) = 12 × 0.35 × 0.65 = 2.73 ✓ Standard deviation = √2.73 = 1.65 students (3 s.f.) ✓
Common mistakes and how to avoid them
Forgetting to check probabilities sum to 1 — Always verify Σp(x) = 1 before proceeding with calculations. This is often needed to find unknown values.
Confusing E(X²) with [E(X)]² — These are different values. E(X²) means square each x first, then multiply by probabilities. [E(X)]² means find the mean, then square it.
Using binomial when conditions aren't met — Don't apply binomial formulas if sampling without replacement from small populations, as probabilities change between trials.
Misreading inequality symbols in binomial problems — P(X ≥ 5) includes 5, but P(X > 5) doesn't. Be precise when using cumulative tables.
Mixing up variance and standard deviation — Remember standard deviation = √variance. Check which one the question asks for.
Rounding too early — Keep at least 4 decimal places in intermediate calculations, especially when calculating variance. Only round your final answer.
Exam technique for "Probability: Distributions"
"Show that" questions — You must show all working steps clearly, even if you know the answer. Write out formulas, substitute values, and arrive at the given answer. Worth 2-3 marks typically.
Command word "Calculate" — Always show your method. Write the formula first, substitute values, then compute. Even if you use a calculator function, show what you're calculating. Usually 2-3 marks depending on steps.
Reading binomial tables — Clearly state what you're reading from the table (e.g., "P(X ≤ 5) = 0.8338 from tables") before performing operations. This demonstrates method even if you misread a value.
Context matters — When writing conclusions about real-world binomial problems, include units and context (e.g., "4.2 students" not just "4.2"). Final answers often require interpretation for the last mark.
Quick revision summary
Probability distributions show all possible outcomes of a random variable with their probabilities (sum = 1). Calculate expected value using E(X) = Σxp(x) and variance using Var(X) = E(X²) - [E(X)]². The binomial distribution B(n,p) models fixed independent trials with constant success probability, giving E(X) = np and Var(X) = np(1-p). Use cumulative binomial tables carefully, noting whether inequalities are strict or inclusive. Always show full working and check your calculations satisfy basic probability rules.