 Papal Probability

This article contained some numbers which led me off on a tangent.

Pope Francis is the 266th pope. There have been 37 false popes.

Therefore, `37/(266+37) = 12.2%` of papal claimants are antipopes.

The average Pope serves for 7.3 years (John Paul II, it turns out, had the second longest reign in history).

Now, the average US life expectancy is 79.8.

Therefore, on average, an American will see `ceiling(79.8/7.3) = ceiling(10.93) = 11` popes in their lifetime.

So, given an antipope rate of 12.2% and 11 popes in a lifetime, what are the odds that you will see an antipope in your life?

My probability is very rusty, and I wasn’t exactly great at it back in school, either, but I’m pretty sure this boils down to a classic Probability Mass Function
``` f(k;n,p) = Pick(n k)(p^k)(1-p)^(n-k) ```
So, the probability of exactly one antipope in a lifetime:
``` f(1;11,.122) = Pick(11 1)(.122)^1(1-.122)^(11-1) = 0.36534 ```

In other words, the odds of you living through exactly 1 antipope are a little over 1 in 3.

But wait, there’s more!

What are the odds of at least 1 antipope in your lifetime? At this point, the maths get pretty long and repetitive, so let’s cheat using the internet:
``` Probability of success on a single trial: 0.122 Number of trials: 11 Number of successes (x): 1 ```