Poisson Distribution: What's the probability of getting a first week without any events when you are told that 5 events occurred within a month? I'm told that the probability of getting $n$ murders per month in London can be modelled as a Poisson distribution with rate $\lambda$. I'd like to calculate the probability that, in a month with 5 murders, there were no murders within the first week. I thought this would simply equal the probability that there are no murders in a week multiplied by the probability that there are 5 murders in the (remaining) three weeks. Is this correct? If not, what should I be doing instead?

Assume a four week month and divide the month into two periods the first week and the remaining three weeks. The Poisson distribution of the number of murders in a period of time is essentially a consequence of the probability model in which the events occur randomly in time. Thus if we have 5 events in the month the probability that they all occur in the second period is $(3/4)^5$. This is the same result as you will get from the conditional probability calculation using Bayes' rule.
