A coping perspective on repeatable rare events

 

The biggest difference between the house and a gambler is their funds: The house can keep playing forever, and a player cannot (or usually does not, at least). The majority of us stops playing simply because we run out of funds, not because we have won enough; therefore the house never loses.

If, however, getting one win is all you want despite vanishingly low odds, how do you decide whether the given game is worth playing?

For me, personally, I like to come up with the number of attempts I need to reach a 50-50 odds. The rest, they say, is up to fate. This is a fairly simple algebraic manipulation:

Let the probability P(success) = s, and the probability P(failure) = f, we can derive the following:

• The probability of one failure:
  
  
  
  
• The probability of k failures in a row:
  
  
  
  
• The probability of at least one success in k attempts:
  
  

  
  

 

Suppose we set our desired probability P(x>=1) at

we have:

Taking the natural log on both sides gives us:

or

The number of attempts needed to reach 50-50 odds is equal to the ln() of those odds divided by the ln() of each attempt's failure probability. To be precise:

The number of attempts = ln(overall failure probability) / ln(a single failure probability)

TODO: Build an interactive web widget for visualizing/calculating the number of attempts that corresponds to a desired overall odds, given an arbitrary binomial success rate.