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The number of miles that a particular car can run before its battery wears out is exponentially distributed with an average of $10,000$ miles. The owner of the car needs to take a $5000$-mile trip. What is the probability that he will be able to complete the trip without having to replace the car battery?

1. $0.5$
2. $0.604$
3. $0.72$
4. None

Exponential distribution is memory less. So how many miles car has already travelled doesn't matter. Let $X$ be the event of car number of miles car travels before dying out.
So mean $\theta=10000$
$\begin{array}{ll} \text{Now } P(X>t) & = e^{-\frac{t}{\theta}} \\ \text{Since }P \bigg( X>\frac{x+y}{X>x} \bigg) &=P(X>y) \\ P(X>5000) &=e^{-\frac{5000}{10000}} \\ & =e^{- \frac{1}{2}} \\ & \approx 0.604 \end{array}$

according to wiki formulae is , then why here it is not divided by mean

We need to use (1- cdf) not pdf in this case P(x>5000) .