Using the binomial distribution

Deciding from a sample

The binomial distribution is not only for computing: it is for deciding. This is where probability meets statistics.

The threshold argument

A company claims 5 % of its parts are defective. A batch of 20 contains 4. Should the claim be believed?

Step 1 — assume the claim is true. If p = 0.05, then X ~ B(20 ; 0.05).

Step 2 — compute the probability of what was observed, or worse.

E(X) = 20 × 0.05 = 1 expected part

P(X ≥ 3) ≈ 0.075        that is 7.5 %
P(X ≥ 4) ≈ 0.016        that is 1.6 %

Step 3 — decide. Observing 4 defects would happen in only 1.6 % of cases if the claim were accurate. That is little: doubting the stated rate is justified.

   probability
       |  █
       |  █  █
       |  █  █  █
       |  █  █  █  █  ▁  ▁          <- observed here: far right
       +--0--1--2--3--4--5---> number of defects
                        ^
              improbable zone if p = 0.05

This reasoning — assume a hypothesis, compute the probability of the observation under it, reject if that probability is too small — is the skeleton of every statistical test. The threshold used (often 5 %) is a convention, not a truth.

What the computation does not say

A small probability proves nothing: it makes the hypothesis implausible. Two errors remain possible:

rejecting a claim that is ACTUALLY TRUE   ->  type I error  (false alarm)
accepting a claim that is ACTUALLY FALSE  ->  type II error (missed defect)

Lowering the threshold reduces false alarms but lets more genuine defects through. No setting removes both at once: it is a trade-off, decided by the cost of each error.

Fields of application

Quality control  :  number of defects in a batch
Surveys          :  number of "yes" answers in a sample
Clinical trials  :  number of patients cured under treatment
Reliability      :  number of components that failed
Genetics         :  number of offspring carrying a trait

In all these cases the model is the same: n independent trials, two outcomes, a constant probability p.

When NOT to use it

This matters as much as knowing how to apply it:

Draw WITHOUT replacement from a small population
   -> p changes at each draw: this is the HYPERGEOMETRIC distribution
   -> tolerable if the sample is < 10 % of the population

DEPENDENT trials
   -> contagion, herd effects, parts from the same misadjusted machine:
      independence fails and the binomial underestimates the risk

VARIABLE probability
   -> a wearing machine sees its p increase over time

Counting over a period, with no fixed number of trials
   -> calls received within an hour: this is the POISSON distribution

The costliest mistake in practice is the third one: assuming independence where there is none. It is what led to underestimating correlated-defect risk in several industrial and financial crises — "improbable" events stop being improbable as soon as they occur together.

Summary