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Apply the binomial distribution to count successes in fixed trials with conditions BINS.
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A random variable follows a binomial distribution if:
Notation: or
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The probability of exactly successes in trials:
where is the binomial coefficient ("n choose k").
When sampling without replacement, trials are not truly independent. However, we can treat them as approximately independent if:
where is the population size. This is the 10% condition.
A basketball player makes 75% of free throws. In 10 attempts, what's the probability of making exactly 8?
When and , the binomial distribution is approximately Normal:
AP Tip: Always verify the BINS conditions before using the binomial distribution. State each condition explicitly on free-response questions.