# Probability of more than 5 responses surveying two locations with different response rates

You are looking for people between 16 to 23 years to answer a survey.
Assume houses on First Street, based on copious amounts of empty beer bottles in front yards and your past experience, has a chance of yielding 16 to 23 year olds to answer your survey of 20% per house.
Houses on Second Street, based upon baby strollers parked all over the place, has a chance of yielding 16 to 23 year olds to answer your survey of 10% per house. It takes 6 minutes to come over to a home, ring a bell, ask if there are any 16 to 23 year olds, and ask the survey question; for simplicity, assume it’s the same amount of time if no one answered the door, or if there are no 16 to 23 year olds.
In one hour, you can survey ten houses. You have an equally efficient partner. You can split up, and survey 10 houses on First Street and 10 houses on Second Street. What is the probability that you get more than 5 survey replies from 16 to 23 year olds in one hour?

How do I go about answering this?

This is not a homework question, it is merely to enhance my understanding of the subject.

## 1 Answer

I think the easiest way is to find the probability of 0, 1, ..., 4 or 5 and then subtract this from 1 (there are lots more ways to get 6 to 20).

$$P(0:5)=\sum_{m=0}^5{\sum_{n=0}^{5-m}{P_1(m)\cdot P_2(n)}}$$

$$P_1(m)=0.2^m \cdot 0.8^{10-m} \cdot {10\choose{m}}$$ $$P_2(m)=0.1^n \cdot 0.9^{10-n} \cdot {10\choose{n}}$$

Doing the arithmatic, I get $1-P(0:5)$ to be about 6.53%