# Poisson Distribution - Conditional Probability 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!

• It is more like the probability of exactly 1 goal is scored in total given that 1 or more goals in total are scored so the expected proportion of those games where goals are scored which have exactly one goal Commented Jan 24, 2023 at 13:52

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 $$\lambda.$$

• Thanks for your answer! Do you mind explaining why P[(N=1) n (N>=1)] = P[N=1]?
– Anna
Commented Jan 24, 2023 at 13:58
• See, Anna, it is the probability of intersection of two events, one being that $N$ is equal to $1$ and other being that $N$ greater than or equal to $1.$ What should be the common event, you think? Commented Jan 24, 2023 at 14:05
• (+1) for the clarity! I wonder if it would be useful to also say something about how to solve (1) in $\lambda$... (the OP seem to be confused about it as well ?) Commented Jan 27, 2023 at 18:06
• I could have @utobi but OP didn't respond further. :-) Commented Jan 27, 2023 at 22:21

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.