# A square is inscribed in a circle. What is the probability that out of 10 points thrown at random independently

A square is inscribed in a circle. What is the probability that out of 10 points thrown at random independently of one another inside the circle, four will fall into a square, three into one segment, and one into the remaining three segments?

1. Let:

R is the radius of the circle;
a – side of the square;
s – area of ​​a circle;
s1 – square area;
s2 – area of ​​one segment.
Then:

a = R√2;
s = πR ^ 2;
s1 = a ^ 2 = 2R ^ 2;
s2 = 1/4 * (s – s1) = 1/4 * (πR ^ 2 – 2R ^ 2) = R ^ 2 (π – 2) / 4.
2. Let’s designate:

p1 = s1 / s = 2R ^ 2 / πR ^ 2 = 2 / π;
p2 = (1 – 2 / π) / 4.
3. Probabilities of events:

a) A – out of 10 points, 4 will fall into a square;

P (A) = C (10, 4) * p1 ^ 4 * (4p2) ^ 6 = 210 * p1 ^ 4 * (4p2) ^ 6;

b) B – out of 6 points, 3 will fall into one segment (there are 4 options for choosing such a segment);

P (B) = 4 * C (6, 3) * p2 ^ 3 * (3p2) ^ 3 = 80 * p2 ^ 3 * (3p2) ^ 3;

c) C – from 3 points – one to the remaining three segments;

P (C) = 2/3 * 1/3 = 2/9.

4. The probability of the event under consideration X is equal to the product of all three probabilities:

P (X) = P (A) * P (B) * P (C);
P (X) = 210 * p1 ^ 4 * (4p2) ^ 6 * 80 * p2 ^ 3 * (3p2) ^ 3 * 2/9 = 11200 * 4 ^ 6 * 3 ^ 2 * p1 ^ 4 * p2 ^ 12 ;
P (X) ≈ 2.1427 * 10 ^ (- 5).
Answer: 2.1427 * 10 ^ (- 5).

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