That's a really big hat...
Number Theory
Level
3
Say you were to put all the positive integers into an infinitely large hat, picking out two of those positive integers at random, α and β. What's the probability that α and β are coprime?
The answer is 0.61.
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1 solution
The probability that a number α is divisible by a prime p is 1/p, since every pth number is divisible by p. Because we're talking about two independent events, the probability that two numbers α and β are both divisible by a prime p is (1/p)(1/p) = (1/p)^2. Thus the probability that α and β are not divisible by a prime p is 1 - (1/p)^2. Now, since our two numbers are coprime, we to have carry this out for all primes. So we end up with (1-(1/2)^2)(1-(1/3)^2)(1-(1/5)^2)... Now the problem relies on the observation that the above expression is the inverse of the Riemann Zeta Function ζ(n) at n = 2. So our answer is 1/ζ(2) = 6/π^2 = .61 to two decimal places.