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3 points are randomly chosen on circumference of the circle. Points are connected to form a triangle. What's the probability that at least on angle will be less than 42 degrees? What's the probability that the sum of any 2 angles is greater than 147 degrees? I would appreciate a solution, but some hints would be fine too, thank you.

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Since the circle is symmetric, we can decide to measure positions around the circle clockwise from the first random points. Then the position of the two other points are still independent and uniformly distributed between $0$ and $360$.

This is useful because our sample space is then a 2-dimensional square and easy to visualize. We can draw a diagram of it with $x$-axis giving the position of the second point and the $y$-axis giving the position of the third point. Then for each of the questions, it is simple enough (though perhaps slightly tedious) to sketch out those areas of the sample space that the event covers and compute their area (they will be polygons with axis-parallel or 45° lines).

Note that "sum of some two angles larger than 147°" is the same as "some angle smaller than 33°", and that an angle in the inscribed triangle is half of the arc spanned by the opposite side.

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  • $\begingroup$ What exactly do you mean by positions of second and third points?(it's he angle from the first point, right?) What inequalities those positions must satisfy so that angles would satisfy the conditions? $\endgroup$ – user1242967 Mar 6 '14 at 20:46
  • $\begingroup$ @user: 1. Yes. 2. That's what I explain in the last paragraph -- the angle at any corner is half the distance along the circle between the two other corners. (This is the Inscribed Angle Theorem). $\endgroup$ – Henning Makholm Mar 6 '14 at 20:51
  • $\begingroup$ So, the inequalities should look like this: y < x + 84 or y < 84 or x < 84 $\endgroup$ – user1242967 Mar 6 '14 at 21:13

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