Q. The amount of time that a watch will run without having to be reset is a random variable having Exponential distribution with mean $\mu=50$ days. Find the probability that such a watch will
(i) have to be reset in less than 20 days.
(ii) not have to be reset in at least 60 days.
(iii) run without a reset an additional 30 days given that the watch has run without a reset for the last 40 days.
My attempt:
Let $t$ be the random variable assumes the amount of time that a watch will run without having to be reset. Given that $t$ has the probability distribution function $$f(t) = \frac{1}{\mu }e^{\frac{-t}{\mu }}= \frac{1}{50 }e^\frac{-t}{50 }$$ Note that $$F(x)=\int_{0}^{x} f(t)dt=1-e^{\frac{-x}{50}}$$ Part i) $P(0 \leq t < 20)=F(20)=1-e^{\frac{-20}{50}}= .3297$
Part ii) $P(t \geq 60)=1-P(0 \leq t < 60)=1-F(60)=e^{\frac{-60}{50}}= .3012$
Part iii) How to deal this conditional probability ? Any thing wrong with
$P(t \geq 30 \mid t \geq 40)=1- P(t < 30 | t \geq 40 )$
$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=1-\frac{P(t<30~ \cap ~t \geq 40)}{P(t\geq 40)}$
$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\frac{P(t\geq 30~ \cap ~t < 40)}{P(t\geq 40)}$
