Paper 2, Section II, D

Probability | Part IA, 2021

Let Δ\Delta be the disc of radius 1 with centre at the origin OO. Let PP be a random point uniformly distributed in Δ\Delta. Let (R,Θ)(R, \Theta) be the polar coordinates of PP. Show that RR and Θ\Theta are independent and find their probability density functions fRf_{R} and fΘf_{\Theta}.

Let A,BA, B and CC be three random points selected independently and uniformly in Δ\Delta. Find the expected area of triangle OABO A B and hence find the probability that CC lies in the interior of triangle OABO A B.

Find the probability that O,A,BO, A, B and CC are the vertices of a convex quadrilateral.

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