Date: Dec 28, 2012 12:28 AM
Author: quasi
Subject: Re: Probability Pill
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(n-1) + 1/n, but

I don't see how to prove it. Can someone provide a proof?

quasi