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Manfred comments on Open thread, July 28 - August 3, 2014 - Less Wrong Discussion
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For a quick reminder of the power of many independent trials, estimate and then answer the following question:
I have 2 biased coins in my pocket. The first comes up heads with probability 51%, while the second comes up heads with probability 49%. I take a coin out of my pocket, uniformly at random, and flip it a million times. I observe that it comes up heads 508,634 times. What is the probability that it is the first coin?
Shortcut: Gurfr ner onfvpnyyl abezny qvfgevohgvbaf jvgu fgnaqneq qrivngvba fdeg(a)/2 = svir uhaqerq (gur snpgbe bs 1/2 pbzrf sebz gur c(1-c) grez va gur inevnapr). Gur qvssrerapr orgjrra gur gjb vavgvnyyl-rdhny cbffvovyvgvrf vf gura nobhg 34 fgnaqneq qrivngvbaf. Jung'f bar zvahf r gb gur zvahf 34 fdhnerq bire 2?