I've been struggling with a problem that was on one of my Math exams several weeks ago. My professor took off several points for my solution, but I'm still not quite sure why it's wrong.
Here's the problem:
Texas has an average of 110 tornadoes per year. What is the probability that the third tornado of the year will happen on the 4th week of the year?
My approach was to calculate the probability of 0, 1, or 2 tornadoes happening in one week using the Poisson distribution. Then I looked at all of the possible combinations of tornadoes over the first three weeks, (for example there could be two in the first week, one in the first, zero in the third, etc.) and made a table of all the possibilities:
I then used basic multiplication and addition rules to find the final probability.
My final answer was 0.031.
My professor argues, however, that you only need to calculate the probability of one tornado occurring in one week using the Poisson distribution, then you use the negative binomial to calculate the probability of the 3rd tornado happening on the 4th week. (with x=4, k=3, and p=probability of one tornado happening in a week)
Her final answer was 0.037. Our answers are very similar, but not equal.
I went over it with my professor, and I'm still not understanding where my process is wrong. The negative binomial assumes a Bernoulli process, which isn't satisfied here as there could be 0, 1, or 2 tornadoes in a given week. Don't you have into account the differing probabilities given by the Poisson distribution?
Any input would be greatly appreciated. I'm really curious to know where I'm going wrong.