# Catching fish in a pond

Suppose the amount of fish in a pond follows a Poisson distribution. A fisherman catches each fish (independently) with probability $\frac{1}{2}$. If $N$ is the total number of fish he catches, what is $\mathbb{P}(N=n)?$ Suppose now that he has caught $N=M$ fish, what is $\mathbb{P}(K=k)$ where $K$ is the number of fish he has not caught?

Say there are $x$ fish in the pond. $\mathbb{P}(N=n)=\frac{1}{2^n}$

$$\mathbb{P}(K=k\,|\,N=M)=\frac{\mathbb{P}(K=k\cap N=M)}{\mathbb{P}(N=M)}=\frac{\mathbb{P}( N=x-k)}{\mathbb{P}(N=M)}=\left(\frac{1}{2}\right)^{x-k-M}$$

This must be wrong because I haven't taken into account the Poisson distribution. How can I incorporate $\mathbb{P}(X=x)=\lambda^x\frac{1}{x!}e^{-\lambda}$ into this ($X$ is the number of fish in the pond)? Help would be appreciated.

• Is the pun "fish following a Poisson distribution" intentional? (Poisson means fish in french). – Denis Feb 10 '14 at 14:34
• @dkuper you know I would always frame Poisson distribution question in term of fishes, there is definitely nothng fishy about this, completely normal. – Lost1 Feb 10 '14 at 14:42
• can you please give us what you attempted? as it stands, i am stuck on proving Rieman's conjecture, because I have never tried it, is that the situation you are in? if so, please have a go at the question yourself. – Lost1 Feb 10 '14 at 14:44
• @Lost1 Do you think you could give me a hint? I am having a lot of trouble with this question... – simon Feb 11 '14 at 9:39
• yes, i actually know the answer to this question, but i have not had time to write it. do you know about moment generating functions? – Lost1 Feb 11 '14 at 12:48

• Thanks for this. Just one thing: why does the second probability not allow for $k<n?$ (the probability that he misses more fish than he has caught) – simon Feb 11 '14 at 15:06