1 out of 1000 boats get hit by lightning each year, if you boat an average amount each year for 20 years what is the probability of getting struck? Having a discussion with my father over this. We found a article from an insurance company that says 1 out of 1000 boats get struck by lightning each year. He says that the probability of getting struck over a 20 year period is 1/50 but I say it is not. Could someone settle our debate and include the math to back it up?
 A: We can assume that your boat getting struck by lightning each year is independent of previous years.
The probability of your boat not getting struck by lightning in a year is $\frac{999}{1000} $, so that the probability of it not getting struck 20 years in a row is
$\left(\frac{999}{1000}\right)^{20} \approx 0.9802 $. Since $1-.9802 = .0198$  there is about 1.98% chance your boat gets struck by lightning in a 20 year period.
A: The probability of being hit ever is $$1-\left(1-\frac1{1000}\right)^{20}$$ You’re father’s calculation is close, but it actually counts the average number of times you’ll be hit in $20$ years.
A very few people will be hit more than once, so the average will be slightly higher than the probability.
$$1-\frac1{1000}$$
is the probability that you won’t be hit in a particular year.
$$\left(1-\frac1{1000}\right)^{20}$$
Is  the probability of not being hit ever, over $20$ years.
So $$1-\left(1-\frac1{1000}\right)^{20}$$
is the probability that you will be hit one or more times in the $20$ years.
That said, given how unlikely the actual probability is exactly $1/1000,$ your dad’s answer is good enough.
