# What is $P(Y=1)$?

Let $$X$$ be the number of tosses of a fair coin required to get the first head. If $$Y | X = n$$ is distributed as Binomial$$(n, \frac{1}{2})$$, then what is $$P(Y = 1)$$?

(A) 4/9
(B) 1/4
(C) 1/3
(D) 5/9


What I understand so far is that $$X$$ follows geometric distribution. The other part, I couldn't relate. Any help would be appreciated.

## 1 Answer

A typical way to solve it would be applying Total Probability Law: $$P(Y=1)=\sum_{n=1}^\infty P(Y=1|X=n)P(X=n)=\sum_{n=1}^\infty n\left(\frac{1}{4}\right)^n$$

• Why multiply by n in the summation? – Muskaan Madan Sep 16 at 12:49
• Try to write $P(Y=1|X=n)$ – gunes Sep 16 at 12:50
• That will be $(1/2)^n$. Did I write correctly? – Muskaan Madan Sep 16 at 12:52
• No, you forget ${n\choose 1}$, check en.wikipedia.org/wiki/Binomial_distribution – gunes Sep 16 at 13:14
• So, the answer would be 1/4? – Muskaan Madan Sep 16 at 15:33