(Sorry for the necroism.)

Gyromaniac, that's not how chance works. 5% chance is 5% every time. Not stacked by the number of tries. Statistics

Actually, Gyromaniac is correct. To fail to have one after 30 tries, at 5% success = 95% failure, you must fail all 30 times, and the probability of that is (0.95) multiplied together 30 times (that is, 30 lots of 0.95 multiplied together, so 29 multiplications), which is 0.2146 (etc.) which is 21.46% chance of the accumulated failure.

The 30th try, however, is still 5% *that*time*.

((Why is it like that?))

The individual attempts are what's called "independent trials with replacement", meaning that no one trial affects the outcome of the next one, and nor is it affected by the previous one ("independent"), while the "with replacement" is more applicable to e.g. drawing cards from a deck where you do or do not put the card you drew back for the next draw. To finish the sequence with no success, you must fail *all* of them. The analysis works like this:

* Fail once = 95% chance.

* Fail twice = 95% chance *if and only if* you already failed once = (95% of 95%) chance from the beginning.

* Fail thrice = 95% chance if and only if you already failed twice = (95% of 95% of 95%) from the beginning.

...

* Fail 30 times = 95% chance if and only if you already failed 29 times = (95% of ... of 95%) from the beginning.