Why was I wrong doing this problem?

Question

"A waste disposal company averages $6.5$ spills of toxic waste per month. Assume spills occur randomly at a uniform rate, and independently of each other, with a negligible chance of $2$ or more occurring at the same time. Find the probability there are $4$ or more spills in a $2$ month period."

The way I did it was to first say "the probability of a spill on a random selected day is $\frac{65}{300}$ (assuming $30$-day months)". Then I calculated the probability that there will be $3$ or $2$ or $1$ or no spills over $600$ days, then I subtracted it from 1. I got a tiny answer (a few percent), but the correct solution was $0.9989$. What did I do wrong, and why is the correct answer $0.9989$?

Answer 1

This sends up all sorts of red flags:

I calculated the probability that there will be 3 or 2 or 1 or no spills over 600 days, then I subtracted 1 from it.

Since probabilities can't be greater than one, if you subtracted one from "the probability that there will be 3 or 2 or 1 or no spills" then you should have gotten a negative number.

Intuitively, you should have realized that if there are an average of 6.5 spills per month, the probability that there will be at least 4 over 2 months should be pretty high, so your low answer should have set off some mental alarms.

Have you learned about the Poisson distribution? That's the easy way to solve this problem.