I calculated the average number of crashes over a 240 week period to be .0875...
what formula do I use to calculate the probability that at least 19 crashes would occur over that 240 week period?
To calculate the probability of at least 19 crashes occurring over a 240 week period, you can use the binomial probability formula. The binomial probability formula is:
P(X ≥ k) = 1 - P(X < k)
where P(X) represents the probability of X occurrences, k is the number of occurrences we are interested in (in this case, at least 19 crashes), and P(X < k) is the probability of X being less than k.
To calculate P(X < k), you need to calculate the cumulative probability up to k-1 occurrences. The formula for P(X < k) is:
P(X < k) = ∑ [nCr * p^r * (1-p)^(n-r)]
where n is the total number of trials (in this case, 240), r is the number of occurrences we are interested in (less than k), nCr is the binomial coefficient calculated as n! / (r!(n-r)!), p is the probability of a single crash (0.0875 in this case), and (1-p) is the probability of no crash.
In your case, you want to calculate P(X < 19), so you would calculate:
P(X < 19) = ∑ [240Cr * 0.0875^r * 0.9125^(240-r)], where r goes from 0 to 18.
Once you have found P(X < 19), you can then calculate P(X ≥ 19) using the formula at the beginning:
P(X ≥ 19) = 1 - P(X < 19)
This will give you the probability that at least 19 crashes would occur over the 240-week period.