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What are cumulative binomial probabilities?
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Example

By some estimates, twenty-percent (20%) of Americans have no health insurance. Randomly sample n = 15 Americans. Let X denote the number in the sample with no health insurance. What is the probability that exactly 3 of the 15 sampled have no health insurance?

Solution. Since n = 15 is small relative to the population of N = 300,000,000 Americans, and all of the other criteria pass muster (two possible outcomes, independent trials, ....), the random variable X can be assumed to follow a binomial distribution with n = 15 and p = 0.20. Using the probability mass function for a binomial random variable, the calculation is then relatively straightforward:

$\binom{n}{k} = \frac{n!}{k!(n-k)!}$

$$P(X=3) = \binom{15}{3}(0.20)^{3}(0.80)^{12} =0.25$$

That is, there is a 25% chance, in sampling 15 random Americans, that we would find exactly 3 that had no health insurance.

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