Poisson Distribution - Conditional Probability Question

Question

I have come across a question in my notes and I do not understand how to solve it. I have looked at the solutions and I still am unclear!

This is the question:

And this is the solution:

I really don't understand what the question is really asking - would anyone be able to explain to me with an example? I also don't understand what the implication of the conditional probability is: e.g. the prob of 1 goal being scored given that 1 or more goals have already been scored? But then that should be independent?

Sorry - again, I am very confused and any help would be much appreciated!

Answer 1

The question has asked, given $N\sim\textrm{Poi}(\lambda), $ to find the value of $\lambda$ such that $\mathbb P[N=1|N\geq 1]= 0.4.$

The solution is nothing but applying the definition of conditional probability and Poisson distribution:

\begin{align}\mathbb P[N=1|N\geq 1]&=\frac{\mathbb P[(N=1)\cap(N\geq 1)]}{\mathbb P[N\geq 1]}\\&= \frac{\mathbb P[N=1]}{\mathbb P[N\geq 1]}\\&= \frac{\frac{e^{-\lambda}\lambda^1}{1!}}{1-\mathbb P[N=0]}\\&=\frac{\lambda e^{-\lambda}}{1-e^{-\lambda}}.\tag 1\label 1\end{align} Equate $\eqref 1$ with $0.4$ and find the value of $\lambda.$

Answer 2

Conditional probability is a measure of a set (N==1) occupies upon the (N>=1) partition (rather than the whole unity, as it is for unconditional probability). Understanding this fact makes conditional probability much more intuitive.

In a physical sense, among all the cases one or more events of interest happened within a given time interval, 40% of the cases only contained one event. Now, the question is, what was the average event rate per time interval (lambda)?

Next, Poisson is a discrete distribution, so we can calculate the probability of N being 1 using Poisson PMF (lambda ^ 1 * exp(-lambda) / 1!) Probability of N>=1 is also easy to be calculated as 1 - PMF(0).

Calculating via 'trial and error' is of course not the best exercise they could think of but at least it's doable.