''Four students, one of whom is a mathematician, takes turns at washing up over a long period of time. The number of plates broken by a given student during this time obeys a Poisson distribution, the probability of any given student breaking $n$ plates being $e^{\lambda }{\lambda }^n/n!$ for some fixed constant $\lambda$, independent of the number of breakages by other students. Given that five plates were broken, find the probability that three or more were broken by the mathematician.''

I decided that the number of plates broken had $\mathrm{Po}(4\lambda )$ and the probability of the mathmo breaking $n$ plates as having $\mathrm{Po}(\lambda )$ as well as the number of plates broken by others having $\mathrm{Po}(3\lambda )$. But I don't know where to proceed, since I've only just learned Poisson distribution.

Hints would be great...

Chris

Now, using your hint about conditional probability, $P(M\ge 3|X=5)$ (calling the number of plates broken $X$ ( $X~\mathrm{Po}(4\lambda )$) $=P(M\ge 3\mathrm{and}X=5)/P(X=5)=e^{4\lambda }{\lambda }^{5}106/120.5!/e^{4\lambda }(4\lambda {)}^5$ which gives the required probability I think.

Many Thanks

Chris