the Bernoulli trial
/ bear-NOO-lee /
A Bernoulli trial is the simplest random experiment imaginable: you do something once and exactly one of two things happens. A coin lands heads or tails; a free throw goes in or misses; a manufactured part is good or defective; a customer either clicks or does not. We agree to call one of the two outcomes 'success' and the other 'failure' — these are just labels, not value judgements, so a 'success' can perfectly well be 'the patient relapses'.
What makes it a trial worth naming is that it carries a single number, p, the probability of success, with the failure probability automatically being 1 - p (often written q). Nothing else is specified — there is no count, no time, no sequence yet. A Bernoulli trial is the atom of probability: it is the one-shot building block out of which most of the named discrete distributions are assembled. Repeat it n times independently and you are heading toward the binomial; repeat it until the first success and you get the geometric; count rare ones over a stretch and you approach the Poisson.
The crucial fine print is the word 'independent' that appears the moment you string trials together: each trial must not be influenced by the others, and p must stay fixed. If you draw cards without replacing them, the probability shifts trial to trial, so those are not identical independent Bernoulli trials — that situation calls for the hypergeometric distribution instead.
Flipping one fair coin once is a Bernoulli trial with success = heads and p = 0.5. Inspecting one item from a line where 2% are defective is a Bernoulli trial with success = defective and p = 0.02.
One action, two outcomes, one number p — the atom that builds the discrete zoo.
'Success' is just a label for the outcome you are counting; it need not be good. The other indispensable assumption when you repeat trials is that p stays constant from trial to trial.