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Topic: Probability Pill
Replies: 20   Last Post: Jan 31, 2013 7:23 PM

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Posts: 575
Registered: 2/10/09
Re: Probability Pill
Posted: Dec 31, 2012 7:16 AM
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On Thursday, December 27, 2012 5:39:14 PM UTC+8, 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.

Let f_{m,n} be the expected value of final number of half pills if
one start with m full and n half pills. f_{m,n} satisfies:

[1a] (m+n) f_{m,n} = m f_{m-1,n+1} + n f_{m,n-1} for m > 0
[1b] f_{0,n} = n

Let f_m(t) be \sum_{n=0..oo} f_{m,n} t^n, [1*] is equivalent to:

[2a] ( m - t + t*(1-t)*d/dt ) f_m(t) = (m/t)*(f_{m-1}(t)-f_{m-1}(0)) for m > 0.
[2b] f_0(t) = t/(1-t)^2

Let u = t/(1-t) and g_m(t) = (1-t) f_m(t), one can simplify [2*] to

[3a] ( m + u*d/du ) g_m(u) = (m/u)*((1+u)*g_{m-1}(u) - g_{m-1}(0)) for m > 0.
[3b] g_0(u) = u

For m > 0, on the L.H.S of [3a],

( m + u*d/du) is an invertible operator over the spaces of polynomials.
furthermore, this operator preserve the degree of a polynomial.

on the R.H.S of [3a],

if g(u) is a polynomial of degree k, so does (m/u) * ((1+u)g(u) - g(0)).

Since [3b] says g_0(u) is a linear polynomial, these observations implies
all g_m(t) are linear polynomials.

Let g_m(u) = a_m + b_m*u and substitute these in [3*], we get:

[4a] a_m = a_{m-1} + b_{m-1}
(m+1) b_m = m b_{m-1}
[4b] a_0 = 0
b_0 = 1

From these, we get

b_m = 1/(m+1)
and a_m = a_0 + \sum_{k=0..m-1} b_{k} = \sum_{k=1..m} 1/m.

So the number we want f_m,0 = f_m(0) = g_m(0) = a_m = 1 + 1/2 + .. + 1/m
as quasi has pointed out in a previous post.

the ( m + u*d/du ) in the L.H.S of [3a] is a invertible operator
over the spaces of polynomials which also preserve the degree of polynomial.
Furthermore, if g(u) is a polynomial of degree k, so does the sort of expression
(m/u) * (( 1 + u)g(u) - g(0) appears in the R.H.S of [3b]. These observations
together with g_0(u) is a polynomial

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