If probability of an event happening is X% and we test it Y times, what is the probability of it happening? The question is pretty much all in the title, but I'll type it out here for extra clarity.
So, I have a function that returns true 10% of the time. I call the function 10 times. If I don't thing too much into it, I would assume the probability of one of them returning true is 10% * 10 (100%). But if you think more about it, that doesn't make sense. Of course there's a probability of all of them returning false.
What is the probability of every function returning false, and how can I calculate it?
 A: If the probability of a single success is $p$ then the probability of "at least one success" in $n$ (independent) trials is
$$p_n=1-(1-p)^n$$
To understand this, think of the complementary event: it consists of all unsuccessful trials. If the trials are independent the probability of this is just the product of the individual failures, which is $(1-p)^n$. Hence you get the result above.
In your example, this gives $p_n = 1 - 0.9^{10}=0.65132156\cdots$
For small $p$, by applying the binomial theorem, one can use the approximation
$$p_n\approx 1 - (1 - np + \cdots) \approx np$$
Only in this case (namely, when $np \ll 1$) the naive approach of summing up the probabilities is (approximately) right.
A: Hint: The probability of the event not happening is the probability of it failing $Y$ times.
A: In order to calculate the probability of at least one successful experiment out of $n$ experiments, you should calculate $1$ minus the probability of the complementary event (i.e., $1$ minus the probability of no successful experiment out of $n$ experiments).
The general formula is $1-(1-p)^{n}$, where $p$ is the probability of success in a single experiment.
In your question $p=0.1$ and $n=10$, hence the answer is $1-(1-0.1)^{10}\approx65.13\%$.
It is worth noting that in order for this method to be correct, the experiments must be independent of each other (i.e., the result of any experiment must not impact the result of any other experiment).
