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?
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asked Feb 13, 2019 at 18:33
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$\begingroup$ 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... . $\endgroup$– whuber ♦Commented Feb 13, 2019 at 18:37
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1 Answer
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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?
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1$\begingroup$ 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. $\endgroup$– TimCommented Feb 13, 2019 at 19:08
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1$\begingroup$ @Tim, you're absolutely right. Edited. $\endgroup$– gunesCommented Feb 13, 2019 at 19:27