Would like to understand the following conditional probability.

Question

I want to understand the following conditional probability and finally how to compute them.

$T$ is an exponential random variable with parameter $\lambda$. For any $s > 0$, compute, $$P(T > t + s | T > s)$$

The density function, $f(x) = e^{-\lambda x}$, is defined where $x>0$ or $0$ otherwise.

Now my intuition, $$P(T > t + s | T > s) = \frac{P(T > t+s \;\cap\; T>s)}{P(T>s)}$$ $$= \frac{1 - P(s \le T \le t+s)}{1-P(T \le s)}\;\;\; (1)$$

I transformed $P(T>t+s\;\cap\;T>s)$ into $1 - P(s \lt T \le t+s)$. Is this correct ? Now in eqaution-$(1)$ I can replace them by the cumulative distribution formula. I need help whether my understanding is correct or not. Moreover should I also consider some bounds on $t$, because the problem does not mention any bounds on it.

Answer 1

It is not true that $P(T>t+s, T>s)=1-P(s<T \le t+s)$ because the complementar of $(T>t+s) \cap (T>s)$ is $(T \le t+s) \cup (T \le s)$.

As @fKonrad says in the comments, use the fact that $(T>t+s) \subseteq (T>t)$.