Question

1. A point is chosen at random in the interior of a circle of radius R. The probability that the point falls inside a given region situated in the interior of the circle is proportional to the area of this region. Find the probability that:

a) The point occurs at a distance less than r (r>R) from the center

b) The smaller angle between a given direction and the line joining the point to the center does not exceed α.

Answer #1

Given that A point is chosen at random in the interior of a circle of radius . Also The probability that the point falls inside a given region situated in the interior of the circle is proportional to the area of this region. See figure below.

a) The probability that the point occurs at a distance less than from the center is

b) The required area is shown below.

The probability that the smaller angle between a given direction and the line joining the point to the center does not exceed is

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