section_path: Applications/Bernoulli trials
section_name: Bernoulli trials
paragraph_text: Bernoulli trials
  
  The number 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 n and plays it n times. As n increases, the probability that gambler will lose all n bets approaches 1/ e. For n = 20, this is already approximately 1/2.789509....
  
  This is an example of a Bernoulli trial process. Each time the gambler plays the slots, there is a one in n chance of winning. Playing n times is modeled by the binomial distribution, which is closely related to the binomial theorem and Pascal's triangle. The probability of winning k times out of n trials is:
  
  In particular, the probability of winning zero times (k = 0) is
  
  The limit of the above expression, as n tends to infinity, is precisely 1/ e.