Editing E (mathematical constant) (section)

=== Bernoulli trials ===
[[File:Bernoulli trial sequence.svg|thumb|300px|Graphs of probability {{mvar|P}} of {{em|not}} observing independent events each of probability {{math|1/''n''}} after {{mvar|n}} Bernoulli trials, and {{math|1 − ''P''&thinsp;}} vs {{mvar|n}}&thinsp;; it can be observed that as {{mvar|n}} increases, the probability of a {{math|1/''n''}}-chance event never appearing after ''n'' tries rapidly {{nowrap|converges to {{math|1/''e''}}.}}]]
The number {{mvar|e}} itself also has applications in [[probability theory]], in a way that is not obviously related to exponential growth. Suppose that a gambler plays a slot machine that pays out with a probability of one in {{mvar|n}} and plays it {{mvar|n}} times. As {{mvar|n}} increases, the probability that gambler will lose all {{mvar|n}} bets approaches {{math|1/''e''}}, which is approximately 36.79%. For {{math|1=''n'' = 20}}, this is already 1/2.789509... (approximately 35.85%).

This is an example of a [[Bernoulli trial]] process. Each time the gambler plays the slots, there is a one in {{mvar|n}} chance of winning. Playing {{mvar|n}} times is modeled by the [[binomial distribution]], which is closely related to the [[binomial theorem]] and [[Pascal's triangle]]. The probability of winning {{mvar|k}} times out of {{mvar|n}} trials is:<ref>{{cite book|first=Mehran |last=Kardar |author-link=Mehran Kardar |title=Statistical Physics of Particles |title-link=Statistical Physics of Particles |year=2007 |publisher=[[Cambridge University Press]] |isbn=978-0-521-87342-0 |oclc=860391091 |page=41}}</ref>
:<math>\Pr[k~\mathrm{wins~of}~n] = \binom{n}{k} \left(\frac{1}{n}\right)^k\left(1 - \frac{1}{n}\right)^{n-k}.</math>

In particular, the probability of winning zero times ({{math|1=''k'' = 0}}) is
:<math>\Pr[0~\mathrm{wins~of}~n] = \left(1 - \frac{1}{n}\right)^{n}.</math>

The limit of the above expression, as {{mvar|n}} tends to infinity, is precisely {{math|1/''e''}}.