A binomial distribution estimates how likely it is to observe k successes in a fixed number n of repeated trials, when each trial has the same success probability p. It is the many-trial cousin of the Bernoulli distribution: instead of asking about one coin toss, ask how many heads appear across ten tosses. section 3.12 section 3.7
Could seven waterproof players be normal variation?
An engineer expects an underwater MP3 player to work 90% of the time. Testing 10 devices produces 7 successes, or 70%. A binomial model asks whether that result is plausible random variation rather than immediate evidence of manufacturing failure. With n=10 and p=0.9, the probability of exactly 7 successes is about 0.0574. section 3.7
Use the model by identifying:
- n: the fixed number of trials. section 3.7
- p: the probability of success on each trial. section 3.7
- k: the number of successes whose probability you want. section 3.12
