Discrete Distributions Binomial and Poisson

The one idea: Most "how many will happen" business questions reduce to one of two counting patterns — a fixed number of yes/no trials (binomial) or an open-ended count of rare events over time or space (Poisson). Recognize which pattern you're in and the probability of any specific outcome — 3 conversions out of 40 visitors, 6 outages in a week — becomes a formula, not a guess.

⚠️ The trap: "We had 8 support tickets yesterday and normally average 5, so something's wrong." Under a Poisson process, day-to-day counts swing a lot on their own — with a mean of 5, getting 8 or more happens more than one day in six just from ordinary randomness. A single count above average is not evidence of a shift; compute the probability of a swing that size under "nothing changed" before you page anyone.

Two kinds of counting questions run the business world. "Out of the 40 leads we called, how many will convert?" is a fixed-trials question — there's a known number of attempts, and each one either succeeds or fails. "How many server outages will we have this week?" is an open-ended question — there's no fixed number of trials, just a rate, and events could in principle happen any number of times. The first pattern is the binomial distribution. The second is the Poisson distribution. Confusing them leads to bad probability estimates and bad alarms; knowing which one you're in turns "how likely is that?" into an actual number.

The binomial distribution: counting successes in a fixed number of trials

The binomial distribution answers: *out of n independent identical…