I often hear people use this as an argument: An event has a 1/X chance of happening in a situation and that situation already happened X times, so the event already happened at least once.
But I don't understand how this is true. If the probability of the event happening is $\frac{1}{n}$, the probability of it not happening is $(1 - \frac{1}{n})$, so the probability of it not happening n times is $(1 - \frac{1}{n})^n$ and the probability of it happening at least once is $1 - (1 - \frac{1}{n})^n$.
Not only this is not equal to $1$ for any $n$, the limit of it as $n$ approaches infinity converges to $1 - \frac{1}{e}$, which is just about 63%.
Your reasoning is exactly correct, assuming the individual trials are independent. The belief that an event with low probability ($1/n$) is bound to occur after $n$ trials is a version of the gambler's fallacy, the belief that there is some kind of compensatory force that increases the probability of the event the longer you've waited without seeing the event. People who believe this may be inclined to justify this by multiplying $1/n$ by $n$ to obtain a value of $1$.
(Of course this compensatory force has to be balanced against the belief in a sustaining force that encourages hot streaks to remain hot. :)