Into the circle K1. of radius R = 1 the square K2 is inscribed, and the circle K3 is inscribed in the square K2.

Into the circle K1. of radius R = 1 the square K2 is inscribed, and the circle K3 is inscribed in the square K2. The probability that a point chosen at random in the circle K1 belongs to the circle K3.

The diameter of the circle K3 is equal to the side of the square d, and the diameter of the circle K1 is equal to the diagonal of the square. Find the diagonal of the square (and the diameter of the circle K1) through the side of the square d:

D = √ (d ^ 2 + d ^ 2) = d√2.

Area of the smaller circle S3:

S1 = (pd ^ 2) / 4

Great circle area S1:

S1 = pD ^ 2/4 = p (d√2) ^ 2/4 = 2pd ^ 2/4 = pd ^ 2/2.

The probability that the point chosen in the circle K1 will belong to the circle K3 is equal to the ratio of the areas S3 to S1:

P = S3 / S1 = (pd ^ 2/4) / (pd ^ 2/2) = 1/2.

Answer: 0.5.