Did you know? One of the easiest ways to lose H1 Mathematics statistics marks is to choose a probability model too quickly. Both binomial and normal questions involve probabilities, but they describe different kinds of random variables.
For JC2 students revising syllabus 8865 in Bukit Timah, the quick test is this: Are you counting successes, measuring a continuous quantity, or working with a sample mean? Answer that before opening the graphing calculator.
Three clues that identify the model
- Binomial B(n, p): a fixed number n of trials; each trial is a success or failure; trials are independent; success probability p stays the same.
- Normal N(μ, σ²): a continuous variable such as a measurement, modelled with a bell-shaped distribution having mean μ and variance σ².
- Sampling distribution: the question concerns a sample mean, not a single observation. Check the population model, sample size and whether a normal model for the mean is justified.
Notice the language. ‘How many customers replied?’ may suggest a count. ‘How long did delivery take?’ suggests a measurement. ‘What is the average time among 64 orders?’ changes the variable again.
Worked check 1: count the successes
A campaign sends 10 independent offers. Each has a 0.30 probability of being accepted. Let X be the number accepted. Then X ~ B(10, 0.30).
The expected number is E(X) = np = 10 × 0.30 = 3. The variance is Var(X) = np(1 − p) = 10 × 0.30 × 0.70 = 2.1.
What is the chance of at least one acceptance? Use the complement: P(X ≥ 1) = 1 − P(X = 0) = 1 − 0.710 ≈ 0.9718.
Here the model does most of the thinking. Once you recognise the event, ‘at least one’ becomes ‘one minus none’.
Worked check 2: measure a continuous result
Suppose a measured delivery time Y is modelled as normal with mean 50 minutes and standard deviation 4 minutes. Write Y ~ N(50, 4²).
To find P(Y < 54), standardise: z = (54 − 50)/4 = 1. Hence P(Y < 54) ≈ 0.8413.
Read the notation carefully: the second parameter in N(μ, σ²) is variance, not standard deviation. Typing 4 when the calculator expects variance, or typing 16 when it expects standard deviation, can change the answer.
Two mistakes worth eliminating
- Do not use ‘normal’ merely because n is large. First identify the variable and the approved syllabus method.
- Do not confuse P(X ≥ 4) with P(X > 4). For a count, those events are different.
One useful syllabus boundary: the 2027 H1 Mathematics 8865 content outline lists binomial and normal distributions but excludes normal approximation to a binomial distribution. Check the paper year before studying techniques that are not tested.
Quick questions
What if the trials are not independent? Do not assume a binomial model. Independence is one of its conditions.
Why does a tutor ask for a sentence before a calculation? Naming the variable and justifying the model can prevent a long, confident solution to the wrong problem.
Explore the Bukit Timah H1 Mathematics probability and statistics overview, or contact eduKate about JC2 revision support and share a recent statistics question that caused difficulty.