Consider a probability distribution on the plane where the probability that a random line passes through a convex region is proportional to its perimeter. Now say there are two equilateral triangles based on opposite sides of a unit square facing away from each other. What is the conditional probability that a random line that passes through one triangle also passes through the other triangle ?
Consider the random process where you keep drawing samples from [0,1] uniformly at random as long as the current sample is larger than the previous sample. What is the Expected value and the Variance of the number of samples you draw.
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