The probability that one or more accidents will occur during any given month is 3/5. The number of accidents that occur in any given month is independent of the number of accidents that occur in all other months. Calculate the probability that there will be at least four months in which no accidents occur before the fourth month in which at least one accident occurs.

I know I should be using the negative binomial distribution to solve this question and that the answer is 0.29. However I don't understand the concept. Can I get an explanation of the answer?

Question

The probability that one or more accidents will occur during any given month is 3/5. The number of accidents that occur in any given month is independent of the number of accidents that occur in all other months. Calculate the probability that there will be at least four months in which no accidents occur before the fourth month in which at least one accident occurs.

I know I should be using the negative binomial distribution to solve this question and that the answer is 0.29. However I don't understand the concept. Can I get an explanation of the answer?

anonymous · Answer

More specifically, I am looking to know why the "r" - the number of desired "successes" is 4 when I do my calculation. If I am only looking to find the probability until at least one accident occurs why does this number not seem to reflect my apparent goal for "successes"? I defined that an accident is a "success" and a month without an accident as a "failure".
The PDF of the Negative binomial distribution is defined as f(x) = (x+r-1)C(r-1) * p^r * q^x.

anonymous · Answer

It seems to me that in this equation, r can stand for more than one thing which is very confusing since they are apparently using the same variable. In one case it seems to represent the current trial number and in another case it seems to represent the total number of trials until the "success".