Q.State, with reasons, which probability distribution — Binomial, Poisson or Normal — is most appropriate for each of the following situations:

Each situation is matched to a distribution by checking whether the variable is discrete or continuous, whether the number of trials is fixed and known, and whether only an average rate is available. …

(i) There is a fixed, known number of trials ($n = 20$ pens), each pen is independently either defective or not (two outcomes), and the probability of a defect is constant at $p = 0.05$ for every pen. All four Bernoulli conditions hold, so this is a Binomial situation. (Since $n$ is reasonably large and $p$ is small, a Poisson approximation with $\lambda = np = 1$ would also be an acceptable close approximation, but the exact, intended model is Binomial.)

(ii) Machine breakdowns per month have no fixed, countable number of 'trials' — only an average monthly rate is known, and breakdowns are described as rare, randomly occurring events over a continuous interval (a month). This is exactly the situation the Poisson distribution is built for. …

A common error is picking Poisson for situation (i) simply because $p = 0.05$ is small, without first checking that $n$ is fixed and known — the defining test for Binomial is the four Bernoulli conditions, not just a small probabili …