Note: The following problem has been mentioned before in another question, but the question related to the problem isn't relevant to mine / it didn't help me.
The number of people that enter a drugstore in a given hour is a Poisson random variable with parameter λ = 10. Compute the conditional probability that at most 3 men entered the drugstore, given that 10 women entered in that hour. What assumptions have you made?
I'm at a complete loss, I'll list the things I've tried down below, but since I'm completely lost, I'm not really able to explain well why I've decided to approach it this way.
#1
P(M $\leq$ 3 | W = 10). Where M = Man, W = Woman.
At first I thought that men and women entering are independent from each other so I did this: $\sum_{a=0}^{3} \frac{P(M = a)P(W = 10)}{P(W = 10)}$
Which isn't correct as they aren't independent from each other at all due to the parameter.
#2
Then I said, well, they aren't independent in that sense, but they are independent when entering individually. There's a 1/2 chance a man enters and 1/2 chance a woman enters. So on average, 5 men enter and 5 women enter. So I set these parameters: $λ_1 = 5; λ_2 = 5$. Where $λ_1$ is the parameter for men and $λ_2$ for women.
And I started doing something similar: $\sum_{a=0}^{3} P(M = a)$ with $P(M=a) = \frac{e^{-5}*5^a}{ a!}$. \
My reasoning behind is, is that it's independent. 10 women entering doesn't change the amount of men entering, because of the change in parameters I did. So I can just calculate the chance of at most 3 men entering in 1 hour.
The answer I got is 0.265. The answer in the solution manual says 0.263 ($39e^{-5}$)
Even though the answer is close in terms of value, I think my reasoning is way off, mainly because I'm not sure how to use the poisson/joint poisson distribution correctly.
So the question is, how do I use the joint poisson distribution correctly in this context, which assumptions do I make and did I do anything correct?