Compute the probability of a uniform and continuous random variable

Question

I have the following problem:

$X$ is a continuous and uniform random variable in $(0, 2\pi)$. Compute the average, the variance and the $P(X> 0.5 \pi | X>0.5)$

I have computed the average ($E[X]= \pi$) and the variance ($Var[X]= \frac{1}{3} \pi^2 = 3.290$). How can I compute the probability requested in the exercise?

Thank you

Answer 1

By the definition of conditional probability,

$P(X > 0.5\pi \mid X> 0.5) = \cfrac{P(X> 0.5\pi \bigcap X> 0.5)}{P(X>0.5)}$

Note that if $X>0.5\pi$ then $X$ must be greater than $0.5$ because $0.5\pi > 0.5$. Thus the above probability becomes

$\cfrac{P(X>0.5\pi)}{P(X>0.5)}$

Do you know how to finish from here?