Q.(from a book)
The six coordinates $(X_i , Y_i ), i = 1,2,3$, of three points $A, B, C$ in the plane are independent $N(0,1)$. Show that the the probability that $C$ lies inside the circle with diameter $AB$ is $1/4$.
I was trying to approach this in terms of area : The probability that C lies inside the circle will be = E(area of circle)/E(total Area). where E is expectation.
I can calculate the numerator by taking expectations and it comes out as pi. I dont know how to calculate the denominator and the intuition for it is also unclear. For ex : if the distribution was say U(-1,1) then denominator makes perfect sense and you can calculate it mentally but for N(0,1) its unclear. Any ideas?