# Probability - Additional mathematical significance to an asymptotic number approaching approximately 0.632? [closed]

This is a math question I have wondered about for around 25 years, and this is my first posting to a math site, so please forgive me if this posting is not up to the typical standards of this site.

Let's look at the set of the probabilities of having at least one occurrence of an event if we make n attempts, where the probability of the event occurring is 1/n. So, for example, if we are looking at the probability of the occurrence of rolling a 4 on a die (1/6), we will actually make 6 attempts. The probability of success in that case (at least one occurrence of a 4 in the 6 attempts) is 1-(5/6^6) = 0.6651 (I have rounded the results.) If we make 15 attempts at something with a probability of its happening (on each separate occasion) being 1/15, then the probability of its occurring at least once in those 15 attempts is 1-(14/15^15) = 0.6447. If we use n=1,000,000, then we get 0.63212. So, we see that we have an asymptotic situation. I have always wondered if there was any other known mathematical significance to this specific approximate number/asymptote. Thanks in advance for reading and for all responses!

Steven

## closed as off-topic by Nate Eldredge, Robert Israel, Carlo Beenakker, Ben McKay, Michael AlbaneseSep 6 '17 at 20:29

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• "MathOverflow is for mathematicians to ask each other questions about their research. See Math.StackExchange to ask general questions in mathematics." – Robert Israel, Ben McKay, Michael Albanese
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• The site you want is math.stackexchange.com; this site is for advanced research-level questions only. – Nate Eldredge Sep 6 '17 at 17:12
• Hint though: it's $1-1/e$. – Nate Eldredge Sep 6 '17 at 17:13
• it's a special case of the limit $\lim_{n\rightarrow\infty}(1-a/n)^n=e^{-a}$ – Carlo Beenakker Sep 6 '17 at 19:33
• Sorry Nate. Thanks for the information. I just posted on math.stackexchange.com. – Steven Herschkowitz Sep 7 '17 at 2:06