Question

There are 3 million people in a region. All of them are "blue" except for 25,000 "red" people. If I was to send a postcard to 1,000 people at random in the region, what are the percentage odds that one of the "red" people would get the postcard?

Answer 1

Technically you should be using a hypergeometric distribution, since you are sampling without replacement. You can make an analogy of this problem with a typical "ball and urn" problem where the hypergeometric is often seen. The problem would be re-worded as: You have 3 million balls in an urn; all are blue except 25,000 are red. You choose 1000 balls at random, what is the probability that 1 of them is red?

If \(X\) represents the number of red people / red balls, then:
\[P(X=1)=\frac{{25,000\choose 1}{3,000,000-25,000\choose 1000-1}}{{3,000,000\choose 1000}} \]
But since the total sample number (3 million) is very large compared to the sample you take (1000), you could probably just use a binomial approximation, and find \(P(Y=1)\) where \(Y\sim Binomial\left(1000,~~\dfrac{25,000}{3,000,000} \right)\)