The Binomial Distribution

Flip a coin 10 times — how likely are exactly 7 heads? A factory ships 100 bolts, each with a 2% defect rate — what's the chance of 3 or more defects? A free-throw shooter hits 80% — in 5 shots, what's the probability she makes all 5? Every one is the same question in disguise: n independent trials, each success-or-failure, count the successes. That shared shape is the binomial distribution.