Probability with multiple uniform distributions

Question

Question:

Two sources output a number at equal rates. The output from source A is uniformly distributed between 100 and 199, and the output from source B is uniformly distributed between 50 and 249. If the number 125 is output, what is the probability that it came from source A?

I'm assuming I need to look at the pdf of each source, but I'm not sure if I have to form some sort of joint probability function. Any help/hints are appreciated.

Answer 1

Hint:

You need to find $P(\text{The number came from source } A|\text{The number is } 125)$. You have $P(\text{The number is }x|\text{The number came from source } A)$ and $P(\text{The number is }x|\text{The number came from source } B)$.

Also I think "Two sources output a number at equal rates." may mean $P(\text{The number came from source } B)=P(\text{The number came from source } A)$.

Now use Bayes rule.

Answer 2

$P\left(A\mid X=125\right)=\dfrac{P\left(A\wedge X=125\right)}{P\left(X=125\right)}=\dfrac{P\left(X=125\mid A\right)P\left(A\right)}{P\left(X=125\mid A\right)P\left(A\right)+P\left(X=125\mid B\right)P\left(B\right)}=...$