Riddler express vitamin solution

I interpreted this as a question about the expectation of a discrete random variable. And I used octave to solve this weeks riddler express.

Suppose a new pill bottle has whole tablets. Each day you pull a tablet, if it's whole, divide it and return the unused half. What's the average number of days until you pull a half tablet?

The outcomes are going to be 1 whole 1 half, 2 wholes 1 half, 3 wholes 1 half. And so on and on until all whole tablets are pulled.

So call each outcome wⁱ h: meaning pull i whole tablets then a half tablet.

For a new bottle w₀ and h₀ are 100 and 0 respectively.

Assume some present state of the bottle w,h. When a whole tablet is pulled, half is consumed and half is returned, so the state becomes w-1, h+1. Which means the bottle always has 100 tablets.

So odds of pulling 1 whole tablet are 100/100, for 2 whole tablets we have 99/100, and so on, note we must take the product of pulling 1 and 2 because the odds of pulling 2 are conditioned on pulling 1 first. This is the falling factorial.

Then we ask, what are the odds of pulling a half tablet after pulling i whole tablets, this is h₀ + i, because pulling i whole tablets means i divided half were placed into the bottle.

We then just solve for the expected value of n.

I got 13.2099606302160.

	% initialize
	w0=100;
	h0=0;
	% sum over i which is the number of whole tablets pulled before the first half
	i=1:w0;
	% sequence of whole tablets
	w = w0-(i-1);
	% number of whole + half in bottle is constant because w-1 + h+1 = w+h
	wh = w0+h0;
	% h is initial plus the number of whole pulled
	h = h0 + i;
	% h denom is the size of the bottle when half is pulled
	hd = w0+h0;
	% nw is probability of pulling n whole tablets in a row
	% it is w0 falling factorial, over bottle size which was shown as constant
	% cumprod is optimization to avoid calculating full falling factorial each term
	nw=cumprod(w./wh);
	% p is proability of n whole tablets then a half tablet
	p=nw.*(h./hd);
	% answer is expection of number of whole tablets pulled
	% so add 1 for the day you pull the half itself
	a=dot(i,p)+1

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