Trial

In statistics a trial is not a court case and not a whole clinical study. It is one repetition of a chance process, and it ends with exactly one outcome from the sample space. Flip a coin once and you have run one trial, landing heads or tails. The problem defines what one trial is, and writing that definition down is the first move, because every count that follows is measured in those units.

Flip a fair coin 20 times and you have run 20 trials. If 12 come up heads, the relative frequency is $12/20 = 0.60$ while the probability of heads is still 0.5 and the expected count was $20(0.5) = 10$. Trials are called independent when the result of one leaves the probabilities for the next unchanged, which fair coin flips satisfy.

Here is the error that quietly ruins simulations. To estimate the chance that a family with three children has at least one girl, one trial is three births, not one birth. "I flipped the coin 100 times, so I ran 100 trials" is wrong twice over: 100 flips supply 33 complete trials with one flip left dangling, and dividing by 100 understates every proportion. A trial runs until the question has a single yes or no answer, so a trial that takes three flips consumes three flips. The exact answer here is $1 - (0.5)^3 = 0.875$, and listing the 8 equally likely birth orders shows 7 of them contain a girl.

The binomial setting demands a fixed number of trials that are independent and share one probability of success. Sampling without replacement breaks the last two: pull chips from a bag and each trial changes what the next one faces, which is why the ten percent condition exists.

Trials are the counting unit of simulation, topic 2.3, Estimating Probabilities Using Simulation.