# Proof of a formula with two geometric random variables

Homework Statement:
If G_1 and G_2 are independent geometric random variables with parameters p_1 and p_2 respectively, show that
Relevant Equations:
P(G_1<G_2)=p_1(1-p_2)/(p_1+p_2-p_1p_2)
The image above is the problem and the image below is the solution I have tried but failed.

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