The Cookie Monster Problem

In general, let there initially be $n$ cookies in each jar. We want to find the probability that, after the Monster discovers that one of the jars is empty, there are $k$ cookies left in the other jar, where $k$ can be any value from $0$ to $n.$ (Note that it could be the case that he has taken the last cookie from each jar, so that when he next reaches into either one of the jars he find that jar to be empty, but in fact both are empty, i.e., $k$ can equal $0$ .)

Now this "event" can happen in one of two ways;

• (i) for the first $2n - k$ cookies taken (and consumed) the Monster has reached into one jar, call it jar $A$ , $n$ times and into the other jar, call it jar $B$ , $n - k$ times. For the $(2n - k + 1)$ th cookie he reaches into jar $A$ , only to find no cookies;

• (ii) same as (i) except for the roles of jars $A$ and $B$ being reversed.

Both scenarios then have a probability of $\dbinom{2n - k}{n} * \dfrac{1}{2^{2n - k}} * \dfrac{1}{2}$ of happening, and so the probability $P(k)$ that the "other" jar has $k$ cookies left in it is

$P(k) = \dbinom{2n - k}{n} * \dfrac{1}{2^{2n - k}}.$

So with $n = 20$ and $k = 0, 1, 2$ we have that

$S = \displaystyle\sum_{k=0}^{2} \dbinom{40 - k}{20} * \dfrac{1}{2^{40 - k}} = 0.372897429....$ ,

and so $\lfloor 1000*S \rfloor = \boxed{372}.$