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 Sep 6 awarded Notable Question Sep 24 awarded Autobiographer Jul 13 accepted Quaternion Group as Permutation Group Jul 13 asked Quaternion Group as Permutation Group Jun 3 awarded Yearling Mar 11 awarded Popular Question Jan 30 comment Nonlinear Second Order Differential Equation Wow. So simple. I feel stupid. Jan 30 accepted Nonlinear Second Order Differential Equation Jan 30 asked Nonlinear Second Order Differential Equation Jan 16 awarded Nice Question Jan 9 comment Taking balls from a bag with replacement Oh, thanks. Never heard of this problem, so it was difficult to google it. Jan 9 asked Taking balls from a bag with replacement Aug 17 awarded Supporter Aug 17 accepted Modern formula for calculating Riemann Zeta Function Aug 17 asked Modern formula for calculating Riemann Zeta Function Dec 30 revised Probability and the Collatz Problem added 561 characters in body Dec 30 comment Probability and the Collatz Problem I would like to offer a thought experiment: If you pick one thousand positive even integers randomly and do one iteration, $C(n)$ of each, then on average, half the results will be even. If you do this again $C^{2}(n)$ then 5/8 will be even, and so on. As you keep going, I think you will find that the average settles at 2/3. I don't think Chebyshev's inequality matters, as that is asking a different question. Dec 30 revised Probability and the Collatz Problem added 1741 characters in body Dec 30 revised Probability and the Collatz Problem edited body Dec 30 comment Probability and the Collatz Problem @Alex Becker I don't think the probability $C^{k}(n)$ is even depends on anything other than the fact that $n$ is even (and positive). The probability tree takes into account all possible paths the Collatz iterations have on $n$, so the probability is just the sum of all paths that end in a positive number, which can be found with a geometric sum.