# Return Period and Probability

My question is simple: If I assume that the probability to be hit by a lightning strike for a person in this year was 0.5 percent would it mean that if I was able to live 200 years I would be hit by one of these at least once?

• If you were hit more than once, then wouldn't somebody else not be hit at all? And maybe that somebody else could be you... .
– whuber
Commented Feb 13, 2019 at 18:37

Of course not. Specifically, the probability of not getting hit at all is (Let $$P(H_i')$$ be the probability of not being hit in day $$i$$):

$$P(H_1')P(H_2')...=P(H_{200}')=\underbrace{(0.995)(0.995)...(0.995)}_{200\ \ \text{times}}=(0.995)^{200}$$ or equivalently $$\left(1-\frac{1}{200}\right)^{200}$$. We multiply the probabilities because these are consecutive events. This expression can be approximated with a well-known identity:

$$\lim_{n\rightarrow\infty}\left(1+\frac{x}{n}\right)^n=e^x$$

Here, our expression is $$\left(1-\frac{1}{200}\right)^{200}=\left(1+\frac{-1}{200}\right)^{200}$$, the expression is not equal but close to $$e^{-1}\approx 0.368$$, which is not that small.

To better grasp the idea, consider fair coins, the probability of (hitting) getting heads is $$0.5$$. Is it guaranteed that you get a head when you toss it twice?

• Since the question was basic, please explain how it was calculated. Also the "which can be approximated as" part may be obvious, but still is worth brief explanation.
– Tim
Commented Feb 13, 2019 at 19:08
• @Tim, you're absolutely right. Edited. Commented Feb 13, 2019 at 19:27