Suppose two coins, $A$ and $B$, are each flipped repeatedly until they land heads. Suppose that the probability that $A$ lands heads each time it is flipped is $\frac14$. Suppose the probability that $B$ lands heads each time it is flipped is $\frac16$. Let $X_A$ denote the number of times $A$ was flipped. Let $X_B$ denote the number of times $B$ was flipped. What is the probability that $X_A=X_B$? (Hint: recall that for any probability $p < 1$, we have $\sum_{i=0}^{\infty}p^i=\frac1{1-p}$).
I tried to use the formula but of no use I tried but I am stuck, thanks a ton for helping!
The probability that $X_A=X_B=n$ is $$\frac14\left(\frac34\right)^{n-1}\frac16\left(\frac56\right)^{n-1}=\frac1{24}\left(\frac{15}{24}\right)^{n-1}$$ since both coins' tosses are independent of each other. It remains to sum these terms for $n$ ranging over the natural numbers, which is where the hint comes in: $$\sum_{n=1}^\infty\frac1{24}\left(\frac{15}{24}\right)^{n-1}=\frac1{24}\sum_{n=0}^\infty\left(\frac{15}{24}\right)^n=\frac1{24}\cdot\frac1{1-15/24}=\frac19$$ Hence $P(X_A=X_B)=\frac19$.