Probability of event when average occurrence rate is known?

Question

If you know that the Empire State Building is struck by lightning an average of 100 times per year then how can you calculate the probability of it being struck at least once within a specified number of days? Would this answer differ from the situation where you truly knew that exactly 100 random events would definitely occur per year? Thanks!

KevinM · Accepted Answer

The way to calculate this is using a POISSON distribution. Even though it might sound fishy - it's the proper way to calculate the probability of a number of events happening in a period of time if you know the RATE at which those events happen:

http://en.wikipedia.org/wiki/Poisson_distribution

So for example - if you know that on average 100 lightning strikes happen every 365 days, then the probability of exactly n strikes in a week would be:

P(n strikes in 7 days) = λ^n e^-λ / n!

where λ is the EXPECTED number of strikes in 7 days:

λ = 100/365 * 7 = 1.918 strikes. So:

P(0 strikes in 7 days) = λ^0 e^-λ / 0! = e^-λ = e^-1.918 = 14.7%

So the probability of 1 or more strikes is:

P(1+ strikes in 7 days) = 1 - 14.7% = 85.3%

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Probability of event when average occurrence rate is known?

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