Toss a fair coin until the last coin toss is different than the previous ones.
Examples :
H, H, H, T
T, T, H
H, T
etc.
What is the probability that the number of tosses, minus 1, is a prime ?
For example : T, T, H we have 2 tails, and 2 is prime. But H, H, H, H, T 4 heads; 4 is not prime.
Without loss of generality lets say first toss is head (H).
Then the sum of the probabilities :
$$S = P(H, H, T) + P (H, H, H, T) + P(H, H, H, H, H, T) + ... $$
$$S = \frac{1}{2^2} + \frac{1}{2^3} + \frac{1}{2^5} + \frac{1}{2^7}... $$
seems to be equal to the infinite sum of the inverses of 2 to the power of the primes.
Obviously this probability is well defined, but what's the value ?
(Approximation for the first thousand primes gives : 0.414682509851)
It's probably irrational, but is it also transcendental ?