Check a binomial model before calculating

Use a binomial model only after identifying a fixed number of trials, two outcomes per trial, independence and a common success probability. The random variable counts successes. A total mass or a waiting time until the first success is a different kind of variable.

Build the method

Sampling without replacement changes the remaining population after each draw. A small sampling fraction can make independence a reasonable approximation. A twenty-percent sample does not meet the usual ten-percent check; a large absolute sample size does not remove this issue.

Worked example

For sixteen independent trials with success probability one half, the expected count is eight. The variance is sixteen times one half times one half, which equals four, and the standard deviation is two successes. The mean, variance and standard deviation are three distinct summaries.

Compare the alternatives

The proportion X/16 has standard deviation 2/16=1/8, while the count X has standard deviation two. Dividing by the sample size changes the units from successes to a fraction. A correct formula with the wrong random variable still answers the wrong question.

Avoid a specific trap

For four fair independent trials, P(X≤1)=P(X=0)+P(X=1)=1/16+4/16=5/16. Exactly one excludes zero, whereas at most one includes it. Listing the allowed counts before using a calculator prevents a common cumulative-probability mistake.

Transfer check with explanation

Transfer: if defect probability doubles after every defect, even a fixed batch size does not make the defect count binomial. Explain which assumptions fail: the next probability changes with prior outcomes, so both constant probability and independence are compromised. Do not merely say the sample is unusual.

WORKED EXAMPLE

A device has defect probability 0.1. Which variable is binomial under the stated assumptions in batch 1?

Check a fixed number of trials, binary outcomes, independence, and constant success probability.
Only the fixed-size independent binary count has all four binomial conditions.
Why an alternative fails: The number of trials is not fixed in advance.

Answer: Defect count among 20 independent devices with the same defect probability

Compare the other answer choices

Number inspected until the first defective device

The number of trials is not fixed in advance.

Total mass of 20 devices

Mass is a quantitative measurement rather than a count of binary successes.

Defect count when each defect doubles the next defect probability

The changing probability and dependence violate binomial assumptions.

Try it, then explain it.

Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.

Practice this skill

Sources & authorship

Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.

AP Statistics Course and Exam Description, effective Fall 2026