Calculating probabilities of events of different time periods The average probability of an event occurring is 3 times in a year.  What is the probability of:
1) an event occurring in any specific month; and
2) 10 events occurring in any specific month?
 A: We use a Poisson model: the number $X$ of events per year has Poisson distribution with parameter $\lambda=3$.
Then it turns out that if $Y$ is the number of events in time interval $t$, then $Y$ has Poisson distribution with parameter $\lambda t$.
Taking a month as $\frac{1}{12}$ of a year, we find that the number of events in a specific month is has Poisson distribution with parameter $\frac{3}{12}$. 
Now to the questions. What does "an event occurring in a specific month" mean? Exactly $1$ event? At least one event? There is lack of clarity.
If it is exactly $1$, then by the usual formula for probabilities governed by the Poisson, the answer is $e^{-3/12}\frac{3/12}{1!}$.
If it is at least $1$, the probability that $Y=0$ is $e^{-3/12}$, so the probability it is $\gt 0$ is $1-e^{-3/12}$.
For the probability of $10$ events in a month, we want $\Pr(Y=10)$. This is $e^{-3/12}\frac{(3/12)^{10}}{10!}$, extremely tiny.
Remark: There are a lot of unreasonable assumptions built into our model. I eat roast turkey on average once a year, roughly as little as I can get away with. The Poisson model would give a very inaccurate result for the probability I eat roast turkey in July.  
