Undergrad Conditional distribution of geometric series

[user158675]

Can someone help me on this question? I'm finding a very strange probability distribution.

Question: Suppose that x_1 and x_2 are independent with x_1 ~ geometric(p) and x_2 ~ geometric (1-p). That's x_1 has geometric distribution with parameter p and x_2 has geometric distribution with parameter 1-p.

Find the distribution of P(x_1 = k| x_1 + x_2 = n)

I found P^-k (1-p)^k-1(2p-1)/1-(1-p)^n-1, but that's certainly wrong.

 

[mathman]

How about let $Q_k=P((x_1=k)\cap (x_2=n-k))=P(x_1=k)P(x_2=n-k)=p^{n-k}(1-p)^k$, then desired prob. $P_k=\frac{Q_k}{\sum_{j=0}^nQ_j}$. The denominator is $P(x_1+x_2=n)$.