A machine fills bottles with a little scatter around 500 ml, while a machine-count of defects is always a whole number. One model uses areas under a smooth curve; the other counts discrete probabilities. Which fits which?
Start with a prediction. State what makes a count binomial, and what a continuous model needs.
Build the model
A binomial distribution 二项分布 needs fixed n, independent trials, two outcomes and a constant p: P(X = k) = C(n,k) pᵏ(1 − p)ⁿ⁻ᵏ, with E(X) = np and Var(X) = np(1 − p). The expected value 期望值 is the probability-weighted mean.
A continuous quantity follows a density curve: probability is AREA, and P(X = x) = 0 for any exact value. The normal distribution 正态分布 standardises with z = (x − μ)/σ before reading a tail probability. A Poisson model counts events at a constant rate, with mean equal to variance.
binomial distribution: The count of successes in n independent two-outcome trials with constant probability p; expected value: The probability-weighted mean of a random variable; normal distribution: The symmetric bell-shaped model for continuous quantities, standardised by z = (x − μ)/σ.
Choose evidence that can test it
Check the binomial conditions before using the formula: sampling without replacement makes trials dependent and changes p, so the model fails. Not every count is binomial.
For a continuous variable, never ask P(X = exactly 130): compute a tail or an interval. State the event before calculating, and draw the shaded region first.
Work from known quantities
State the known values and their units. Choose the relation because its assumptions fit this case, then rearrange before substitution.
Known: X ~ binomial(5, 0.2). P(X = 0) = 0.8⁵ = 0.32768. E(X) = np = 1 and Var(X) = np(1 − p) = 0.8. For a normal quantity with μ = 100, σ = 15, the value 130 has z = (130 − 100)/15 = 2, so it lies two standard deviations above the mean.
Example:
For X ~ binomial(5, 0.2), find P(X = 1) using the same sequence: choose the model → formula → substitution. (Answer: P(X = 1) = C(5,1) × 0.2 × 0.8⁴ = 5 × 0.2 × 0.4096 = 0.4096.)
Check the conclusion and its limits
Binomial moments follow only when all four conditions hold. A normal model needs roughly symmetric data; a continuity correction matters when approximating a discrete distribution by a normal one.
Return to the original observation. Explain what the result supports, which conditions it assumes, and one way to test a competing explanation.
Warn:
Every count of successes has a binomial distribution regardless of dependence. This claim is false: the model requires independent trials with constant p — sampling without replacement or a changing success rate breaks it, so the formula would report a false precision.
Key:
Statistical distributions: binomial for fixed-n independent counts (E = np, Var = np(1 − p)); normal for continuous quantities via z = (x − μ)/σ; probabilities are areas for continuous models and exact terms for discrete ones.
