TITLE: 63% chance of event happening over repeated attempts
QUESTION [5 upvotes]: This question is of interest:

If there is a $1 / x$ chance of something happening, in $x$ attempts, for
  large numbers over $50$ or so, the likelihood of it happening is about
  $63\%$. If there's a $1$ in $10\,000$ chance of getting hit by a meteor if you
  go outside, if you go outside $10\,000$ times, you have a $63\%$ chance of
  getting hit with a meteor at some point. If there's a $1$ in a million
  chance of winning the lottery and you buy a million (random) lottery
  tickets, you have a $63\%$ chance of winning.

What is the main idea underlying this property?

REPLY [2 votes]: It's a limit problem.. the limit of 1-(x-1)/x)x as x approaches infinity is 1-(1/e) which is ~ 63%.
So yes, if you have a one in a million chance of winning the lotto, and you buy 1 million random tickets, you have a 63% chance of winning.