Problem: Suppose the number of cars passing a road follows Poisson distribution with average 5 cars per minute. Also suppose the percent of red cars follows Poisson distribution with average 20% cars being red.
Then how would you answer something like: what is the probability of at least 50% of the cars passing the road in 10 minutes are red?
Approach: I can do this problem brute force:
- Average number of cars expected to see in 10 minutes is 50 cars.
- Then from there calculate probability of seeing 1 car, 2 cars, all the way up to a large number of cars -until probability ends up being close to 0.
- Then for X given number of cars, determine the probability >50% are red. Multiple this probability by the probability of seeing X number of cars. Repeat this for all number of cars and add up the probabilities.
Is there a faster / more methodical way of answering this question?