Probability - Bus stop and coffee shop

Question

I'm having a bit of trouble with the following question. I can do the previous parts where you are calculating expectation values for your length of walk etc but I am not sure on the below. Does it have some conditional element eg p(you can fetch a coffee | you have picked the closest route to the bus stop)

Any help would be great. Thanks.

Q) You arrive at a road and all you know is that you are somewhere between two bus stops which are one mile apart.

You also know that there is a coffee stop somewhere between the two bus stops.

If you head to the closer bus stop, what is the chance that you can fetch a coffee on your way?

Original question attached for full context

![1]: https://i.stack.imgur.com/bcvYO.png

Answer 1

Hint: If $Y$ is the position of the coffeeshop and $X$ is your position, then you will walk past the coffeeshop if and only if $Y \le X \le \frac{1}{2}$ or $\frac{1}{2} \le X \le Y$. Here, $X$ and $Y$ are indepentent random variables, distributed uniformly on $[0, 1]$.

Answer 2

Hint: Since you don’t know where you are, you assume that your position $X$ is a uniform $[0,1]$ random variable. You can just use the properties of uniform random variables, which makes it a lot easier to solve.