quasi
Posts:
12,054
Registered:
7/15/05


Re: Probability Pill
Posted:
Dec 28, 2012 12:28 AM


quasi wrote: >quasi wrote: >>quasi wrote: >>>quasi wrote: >>>>William Elliot wrote: >>>> >>>>>Each day I take 1/2 an aspirin tablet. I bought a bottle >>>>>of 100 tablets; each day I take out one, if it's whole I >>>>>break it half and eat a half and put the other half back: >>>>>if I pull out a half tablet I eat it. I was wondering >>>>>after I break the last whole one what the expected number >>>>>of halves are in the bottle? I assume that any piece I >>>>>pull out has uniform probability. >>>> >>>>I suspect that the above question is not actually yours. >>>> >>>>If that's the case, what is the actual source? >>>> >>>>Is it from a poster in another forum? If so, why do you omit >>>>mention of the poster and the forum? >>>> >>>>Is it from a book or math contest? >>>> >>>>Why do you repeatedly post questions that are not your own >>>>without giving credit to the source? >>>> >>>>In any case, the expected number of halves left when the last >>>>whole pill is split is >>>> >>>> (199!) / ((4^99)*((99!)^2)) >>> >>>Which is slightly more than 11 half pills. >> >>Oops  ignore my answer  it's blatantly wrong. >> >>I'll rethink it. >> >>In the meantime, can you identify the source of the >>problem? > >Ok, the correct answer is x/y where x,y are given by > > x = 14466636279520351160221518043104131447711 > > y = 2788815009188499086581352357412492142272 > >As a decimal, x/y is approximately 5.18737751763962 > >Thus, on average, about 5 half pills.
As a point of interest, when starting with n pills, the expected number of half pills remaining at the end (after the last whole pill is gone), as confirmed by the data, is
1 + 1/2 + 1/3 + ... + 1/n
However, while I'm sure the above result is correct, I don't have a proof.
Let f(n) denote the expected number of half pills remaining at the end, starting with n whole pills at the outset. It's immediate that f(1) = 1, hence a natural approach would be to try to prove the recursive relation f(n) = f(n1) + 1/n, but I don't see how to prove it. Can someone provide a proof?
quasi

