recognizing which discrete model a problem calls for

Word problems rarely say 'use the binomial'. Choosing the right discrete distribution is a skill of asking a few diagnostic questions about the structure of the situation, and it is what turns a list of formulas into a usable toolkit. The single most useful question is: what exactly is the random quantity I am being asked about — a count of successes, a number of trials I wait through, or something else?

Here is a decision path. Is it a single yes/no outcome? That is Bernoulli. Are you counting successes in a fixed number n of independent trials with a constant success probability? Binomial — unless you are drawing without replacement from a small finite pool, in which case it is hypergeometric. Are you counting how many trials until the first success? Geometric. Until the r-th success? Negative binomial. Are you counting how many rare events happen in a fixed window of time or space at a steady average rate, with no natural upper limit? Poisson. Are all finitely many outcomes equally likely? Discrete uniform. Are you splitting trials among more than two categories and counting each? Multinomial. Two pivot questions cut through most cases: (1) is the number of trials fixed in advance, or are you waiting for a target number of successes? Fixed trials point to binomial or hypergeometric; waiting points to geometric or negative binomial. (2) When sampling, do you replace items or not? Replacement keeps p fixed (binomial); no replacement makes p drift (hypergeometric).

Then sanity-check by matching features. The Poisson alone has mean equal to variance; if your data's variance far exceeds its mean, suspect overdispersion and reach for the negative binomial or beta-binomial instead. The biggest real-world trap is silently assuming independence and a constant probability when neither holds — contagious or clustered events break the Poisson, and small finite pools break the binomial. Naming the distribution is the easy half; checking that its assumptions actually fit your problem is the half that keeps you honest.