A problem by Charles Lien
Two points are in a plane at a distance of 2 units from each other. At time t=0, they start moving at velocity v = 1 (unit per second) in random directions with uniform probability distributions. Let P be the probability that there is a point in time in which the distance between the two points is less than or equal to 1. What is the greatest integer that is less than or equal to 1000*P?
The answer is 166.
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1 solution
Using the principle of superposition, we see that if we set one of the two points to be our reference point, then the other point will move with a random speed between 0 and 2 (not with uniform probability distribution) with uniform probability distribution for the angle. The two points will come within 1 unit of each other if the angle that the second point moves is between sin − 1 ( 2 1 ) and sin − 1 ( 2 1 ) , which is a probability of 6 1 . ⌊ 1 0 0 0 ⋅ 6 1 ⌋ = 1 6 6 .