Exponential Distribution conditional probability $P(X<x+t\mid X>x)$

Question

So i know that the exponential Distribution is memoryless so $P(X>x+t\mid X>x)=P(X>t)$. However, when we have $P(X<x+t\mid X>x)$ we cannot apply this right? So we have to use that definition of the conditional to get $\frac{P(x<X<x+t)}{P(X>x)}$. For example we want the first goal in a game to happen in time $x+t$ and we know that it did not happen in the first $x$ time. We would apply what i wrote out to calculate the probability of that, right?

Thanks

Answer 1

$$ \Pr(X<x+t\mid X>x) = 1 - \Pr(X>x+t\mid X>x). $$

Answer 2

The Exponential cumulative distribution function is so easy-going, that applying Bayes rule to find the conditional statement is trivial - and one will again get lack of memory,

$$P(X<x+t \mid X>x) = P(X<t)$$