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An experimental drug is being tested for side effects. The producers of the drug claim that only 5% of users will experience side effects. If 20 people are tested what is the probability that 2 or more will experience side effects?

Answer: 0.2641

How do you do this problem? Looking at the Probability Wikibook didn't help me..

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It's out of this Barron's Fundamentals of Engineering (FE) Exam book. I looked at the reference book and in this Barron's book too, but they don't give much explanation. – user8731 Mar 27 '11 at 0:42

You assume that the 20 tests are independent, which allows you to use a binomial distribution. The probability that $k$ tested persons experience side effects is then:

$$\mathbb{P}(X=k)={20\choose k} \cdot 0.05^k \cdot 0.95^{20-k} \; .$$

Then, if you want the probability that at least two people experience side effects, you compute the probability that at most $1$ will and take the complement:

$$\mathbb{P}(X\geq 2) = 1- \mathbb{P}(X < 2) = 1 - \mathbb{P}(X = 0) - \mathbb{P}(X = 1) \; .$$

I'll let you fill in the details.

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Thank you! This pointed me in the right direction: I read about uniform and binomial distributions, tracked it down in my FE reference handbook, and learned how to use nCr on my TI-30x :) – user8731 Mar 27 '11 at 1:54

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