Imagine I have two people, L and J shooting to a target.
Both shoot 5 times, where the probability for L to hit the target is $\frac 1 2$ and the probability for J is $\frac 1 4$. Both independent.
Now, knowing that the target was hit 5 times total, what is the probability that exactly 2 hits were made by L.
I solved it with conditionals, and it was ok.
If I define L:#hits made by L and J:#hits made by J. I know that both are independent binomial variables: $L\sim \mathcal{Bin}(5,1/2)$ and $J\sim\mathcal{Bin}(5,1/4)$.
So, if I call $S = L + J$ I can calculate: $\Pr(L = 2 \mid S = 5) = \frac{\Pr(L = 2, S = 5) }{ \Pr (S = 5)} = \frac{[\Pr(L = 2) \cdot \Pr(J = 3)]}{\Pr(S = 5)}$ where $\Pr(S = 5)$ can be calculated with the total probability theorem.
Now, I can solve this and is okay, but then I have to continue with other examples and I have to recalculate everything everytime.
I was told that if i calculate the variable $L\mid S$, I obtain a new binomial variable with: $p= \frac{P_l}{(P_l + P_j)} = \frac{(1/2)}{(1/2 + 1/4)}$.
I cannot get to that, what should I do?