another exponential distribution probability example

Question

I have an exponential variable with $\mu = 169$ hours which represents time to failure for a part in a machine. I know that the probability that it will last for 100 hours or less is about 0.45.

Given that the part has been used for 100 hours, what is the probability that it will last for 100 more? I think the memoryless principle applies here, but I can't figure out how.

Answer 1

Use Bayes' rule:

$$P(t>200 | t>100) = \frac{P(t>200) P(t>100 | t>200)}{P(t>100)}$$

where I think the notation is rather straightforward and of course $P(t>100 | t>200) = 1$.

Perhaps this figure will help:

Answer 2

Let $X$ be an exponential random variable. The memoryless property says that $$ P(X > t+s | X >s) = P(X>t) $$ for any $t\geq 0$ and $s \geq 0$. Recall that the vertical bar "$|$" in the probability statement here can be read as "given" or "conditional on". In your question, we have both $t=100$ and $s=100$. Can you figure it out from here?