1994 Paper 3 Q14

Year: 1994
Paper: 3
Question Number: 14

Course: LFM Stats And Pure
Section: Geometric Probability

Difficulty: 1700.0 Banger: 1516.0

probability geometric probability circle triangle inclusion-exclusion inequality uniform distribution

Problem

Three points, \(P,Q\) and \(R\), are independently randomly chosen on the perimeter of a circle. Prove that the probability that at least one of the angles of the triangle \(PQR\) will exceed \(k\pi\) is \(3(1-k)^{2}\) if \(\frac{1}{2}\leqslant k\leqslant1.\) Find the probability if \(\frac{1}{3}\leqslant k\leqslant\frac{1}{2}.\)

No solution available for this problem.

Rating Information

Difficulty Rating: 1700.0

Difficulty Comparisons: 0

Banger Rating: 1516.0

Banger Comparisons: 1

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Problem source

Three points, $P,Q$ and $R$, are independently randomly chosen on the perimeter of a circle. Prove that the probability that at least one of the angles of the triangle $PQR$ will exceed $k\pi$ is $3(1-k)^{2}$ if $\frac{1}{2}\leqslant k\leqslant1.$ Find the probability if $\frac{1}{3}\leqslant k\leqslant\frac{1}{2}.$

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