# Primes and the Geometric Distribution

by CTS
Tags: distribution, geometric, primes
 P: 20 Given the probability of flipping a heads with a fair coin is $$\frac{1}{2}$$, what is the probability that the first heads occurs on a prime number?
 Sci Advisor PF Gold P: 2,226 $$\sum_{n=1}^\infty \left(\frac{1}{2}\right)^{p_n}$$ where $p_n$ is the nth prime number. Which gives a value of about 0.41468
 Sci Advisor HW Helper P: 2,586 Let {$p_i$} be the sequence of primes. The probability you're looking for would be: $$\sum _{i=1} ^{\infty} 0.5^{p_i}$$
 P: 20 Primes and the Geometric Distribution Yes, I realize that the answer is this summation ($$\sum _{primes} ^{} 1/2^p$$), which clearly converges very quickly (to about .4146825...). Anyone have any ideas on whether the answer is irrational, or even expressable as a fraction of constants (like $$\sum _{n=1} ^{\infty} 1/n^2$$. Anyone know anything else about $$f(x)=\sum _{primes} ^{} 1/x^p$$?
 Emeritus Sci Advisor PF Gold P: 11,155 If you had posted this question several centuries ago, I might have said: "the sum approaches $$\sqrt{2} - 1$$ but there is not enough room in this forum for me to prove it. " and gotten away with that !