Dinosaur

01-11-07, 01:33 PM

If the following came from a less credible source, I would suspect that there is no unambigous solution.Four points are chosen uniformly and independently at random in the interior of a given circle. Find the probability that they are the vertices of a convex quadrilateral.This problem is from a competition for college mathematics majors. Unfortunately, it was sent to me without any clue as to how to solve it.

Can anyone here come up with a solution to the above?

After ignoring them for a day or so, I have often come up with solutions to problems which seemed intractable when first encountered. I do not expect to come up with a solution to the above.

In spite of the source, I am not 100% certain that it is an unambiguous problem statement. Random is often an ambiguous term without some context. In general, processes are random, but numbers and geometric objects are not.

The following is an example of an ambiguous geometric probability problem.What is the probability that a random chord of a circle will be longer than the length of an inscribed equilateral triangle?The following are three random processes leading to the construction of a chord.Pick an arbitrary point on the circumference and draw a tangent to the circle at that point. Generate a chord by spinning a straight line about that point. If the angle between the line and the tangent is 60 to 120 degrees, the chord will be longer than the side of an inscribed equilateral triangle. For angles 0 to 60 and 120 to 180, the chord will be shorter. Hence probability is 1/3

Inscribe a circle interior to an equilateral triangle and tangent to the three sides (The triangle is inscribed in a larger circle). Choose a random point in the larger circle as the midpoint of the chord. If the point is interior to the smaller circle, the chord will be longer than the side of the equilateral triangle. The area of the inner circle is 1/4 the area of the larger circle. Hence, the probability is 1/4.

Pick a diameter of the circle. Choose a random point on that diameter as the midpoint of a chord perpendicular to the diameter. If the point is in the middle half of the diameter, the chord will be longer than the side of an inscribed equilateral triangle. Hence the probability is 1/2The above indicates a potential problem with geometric probability problems. There is no paradox, merely three different results depending on which random process is used to generate a random chord. For the above problem random chord is an ambiguous term.

Can anyone here come up with a solution to the above?

After ignoring them for a day or so, I have often come up with solutions to problems which seemed intractable when first encountered. I do not expect to come up with a solution to the above.

In spite of the source, I am not 100% certain that it is an unambiguous problem statement. Random is often an ambiguous term without some context. In general, processes are random, but numbers and geometric objects are not.

The following is an example of an ambiguous geometric probability problem.What is the probability that a random chord of a circle will be longer than the length of an inscribed equilateral triangle?The following are three random processes leading to the construction of a chord.Pick an arbitrary point on the circumference and draw a tangent to the circle at that point. Generate a chord by spinning a straight line about that point. If the angle between the line and the tangent is 60 to 120 degrees, the chord will be longer than the side of an inscribed equilateral triangle. For angles 0 to 60 and 120 to 180, the chord will be shorter. Hence probability is 1/3

Inscribe a circle interior to an equilateral triangle and tangent to the three sides (The triangle is inscribed in a larger circle). Choose a random point in the larger circle as the midpoint of the chord. If the point is interior to the smaller circle, the chord will be longer than the side of the equilateral triangle. The area of the inner circle is 1/4 the area of the larger circle. Hence, the probability is 1/4.

Pick a diameter of the circle. Choose a random point on that diameter as the midpoint of a chord perpendicular to the diameter. If the point is in the middle half of the diameter, the chord will be longer than the side of an inscribed equilateral triangle. Hence the probability is 1/2The above indicates a potential problem with geometric probability problems. There is no paradox, merely three different results depending on which random process is used to generate a random chord. For the above problem random chord is an ambiguous term.