320 6 Combinatorics and Probability
930.Two airplanes are supposed to park at the same gate of a concourse. The arrival
times of the airplanes are independent and randomly distributed throughout the 24
hours of the day. What is the probability that both can park at the gate, provided
that the first to arrive will stay for a period of two hours, while the second can wait
behind it for a period of one hour?
931.What is the probability that three points selected at random on a circle lie on a
semicircle?
932.Letn≥4 be given, and suppose that the pointsP 1 ,P 2 ,...,Pnare randomly chosen
on a circle. Consider the convexn-gon whose vertices are these points. What is
the probability that at least one of the vertex angles of this polygon is acute?
933.LetCbe the unit circlex^2 +y^2 = 1. A pointpis chosen randomly on the
circumference ofCand another pointqis chosen randomly from the interior ofC
(these points are chosen independently and uniformly over their domains). LetR
be the rectangle with sides parallel to thex- andy-axes with diagonalpq. What is
the probability that no point ofRlies outside ofC?
934.If a needle of length 1 is dropped at random on a surface ruled with parallel lines at
distance 2 apart, what is the probability that the needle will cross one of the lines?
935.Four points are chosen uniformly and independently at random in the interior of a
given circle. Find the probability that they are the vertices of a convex quadrilateral.