# Probability that a sample represents between X% and Y% of the population

In a country that has a population between 1500000 and 3000000 people, there are 2262 legal first names and 1030 legal last names. A statistician picks a sample of 225 000 people to try to compute the name distribution. However, he is informed that although the name distribution is unknown, there is necessarily two people with the same First name + last name combination.

What is the probability that the sample represents between 7.4% and 8.6% ?

The fact that there must be a duplicate first name/last name pair means there are at least ???? people in the population. For how big a population would $225,000$ be $7.4\%$? For how big a population would $225,000$ be $8.6\%$? I think you are supposed to find the interval of population that meets the criteria, assume that all values are equally likely (!?!?) and find what fraction of the values your sample is within the range $7.4\%$ to $8.6\%$. A badly worded question.
• Yes, that is what I was saying. We have now reduced the range of possible populations to $[2329861,3000000]$. This is the interval I was talking about. – Ross Millikan Jan 26 '16 at 16:51