The binomial distribution models the number of successes in a fixed number of independent trials when each trial has only two possible outcomes, usually labeled success and failure, and the probability of success remains constant from trial to trial. These four conditions define the binomial setting: fixed number of trials, independent trials, two outcomes per trial, and constant success probability. Examples include the number of heads in 10 coin flips, the number of defective items in a sample when the defect probability is constant, or the number of students who pass an exam out of a fixed group. The normal distribution describes continuous bell-shaped data, not counts of successes. The Poisson distribution models counts of events occurring over a fixed interval when events occur at a constant average rate. The uniform distribution assigns equal probability across outcomes or intervals. Because the question explicitly states “number of successes” and “fixed independent trials,” the correct model is binomial. Study Guide references/topics: binomial distribution, independent trials, success probability, discrete random variables.

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