Rains On Average once a week chances of it raining 3 times in 4 week

Question

A friend of mine asked me this

It rains On Average once a week. What are the chances of it raining 3 times in 4 week

And I was wondering how to approach this.

Thank you.

Answer 1

Let $X$ be # of rains in a week. If the average # of rains per week = $1$, $X \sim \mathrm{Pois}(\lambda = 1)$ and by dimensional analysis (unit coversion):

$$\lambda = \cfrac{1 \text{ rain}}{1 \text{ week}} = \cfrac{4}{4} \ \cdot \ \cfrac{1 \text{ rain}}{1 \text{ week}} = \cfrac{4 \text{ rain}}{4 \text{ week}}$$

This gives us the average # of rains in $4$ weeks. So if $Y$ = # of rains in $4$ weeks, $Y \sim \mathrm{Poisson}(\lambda = 4)$, and the question asks $P(Y=3)$.