Why is $0.63212$ the probability of a $\frac1n$-probability event happening in $n$ trials?

Question

I've always assumed on faulty intuition that if you have an event which occurs 1 in n chances, it will be super likely to happen at some point of that event occuring n times. However, given some analysis, it doesn't actually seem to be all that super likely, and seems to converge at a particular value as the value of n rises. That value is about 0.63212.

Is this correct? If so, is there a name for this value and is it considered significant within the field of probability?

Below is the Python code that I used to arrive at this value.

>>> def p(x, r):
...   return x + r * (1.0 - x)

>>> def p_of_1(r):
...   x = r
...   while True:
...     yield x
...     x = p(x, r)

>>> def p_of_n(n):
...   g = p_of_1(1.0 / n)
...   return [next(g) for x in range(n)]
...

>>> p_of_n(1)
[1.0]
>>> p_of_n(2)
[0.5, 0.75]
>>> p_of_n(3)
[0.3333333333333333, 0.5555555555555556, 0.7037037037037037]
>>> p_of_n(4)
[0.25, 0.4375, 0.578125, 0.68359375]
>>> p_of_n(5)
[0.2, 0.36000000000000004, 0.488, 0.5904, 0.67232]

>>> p_of_n(6)[-1]
0.6651020233196159
>>> p_of_n(10)[-1]
0.6513215599000001
>>> p_of_n(100)[-1]
0.6339676587267709
>>> p_of_n(10000)[-1]
0.6321389535670703
>>> p_of_n(10000000)[-1]
0.6321205772225762

Answer 1

It's easier to work backwards. The probability that the event does not occur on a single try is, of course, $1-\frac 1n$. It follows that the probability that it fails to occur in $n$ trials is $p_n=\left(1-\frac 1n\right)^n$. Therefore the probability that it occurs at least once in those $n$ trials is $$1-p_n=1-\left(1-\frac 1n\right)^n$$ If we now recall the limit definition of the exponential: $$e^a=\lim_{n\to \infty}\left(1+\frac an\right)^n$$ We see that, for large $n$, $$1-p_n\sim 1-\frac 1e=0.632120559\dots$$