# probability – Expectation of a conditioned geometric random variable

So first let $$X$$ be the number of independent coin tosses until first head; $$P(H)=p$$. Suppose we have observed that the first coin toss is tails, what is the expectation $$E(X-1|X>1)$$?

So intuitively this is just equal to $$E(X)$$ which makes sense to me. I tried to derive this more formally:

$$E(X-1|X>1)=sum_xg(x)p_{X-1|X>1}(x)=sum_x(x-1)p_{X-1|X>1}(x)$$

Since a geometric random variable is memoryless, we have:

$$E(X-1|X>1)=sum_x(x-1)p_{x}(x)=E(X)-1$$

I seem to be off by a factor of 1, I suspect it might be something to do with the fact that I am still summing over all $$x$$ as denoted in $$sum_x$$ but I am not quite sure how to fix this.