Probability That Random Chord Exceeds Radius in a CircleDate: 11/20/2004 at 12:25:45 From: Thomas Subject: Probability of a circle (some geometry involved) If a chord is selected at random on a fixed circle, what is the probability that its length exceeds the radius of the circle? I understand the basic concept, but I can't pin down how to do it. All I have figured out is that if you drew the line straight through the center, then it would be twice the radius. Date: 11/20/2004 at 18:26:35 From: Doctor Anthony Subject: Re: Probability of a circle (some geometry involved) Hi Thomas - You can get 3 different answers depending on how you define the random chord. Method (1) ---------- You get a random chord by choosing a point at random in the circle and letting this point be the midpoint of the random chord. The distance from the centre of the circle to a side of an inscribed regular hexagon (whose sides are equal to the radius) is a*cos(30) = a*sqrt(3)/2 where a = radius of the circle. The area of the circle within this distance from the centre is therefore pi*(3a^2/4) So the probability that a random chord exceeds the radius of the circle is this area divided by the area of the full circle pi*a^2 and it follows that the probability is (pi*3a^2/4)/(pi*a^2) = 3/4. Method (2) ---------- You draw a chord prallel to a given line with the position of the chord uniformly distributed along a diameter which is perpendicular to the given line. The distance of the mid-point of a side of the inscribed hexagon from the centre of the circle is a*cos(30) = a*sqrt(3)/2. If x = distance of mid-point of chord from the centre, then the length of the chord is greater than the radius of the circle if x < a*sqrt(3)/2. So the probability that a chord is greater than the radius of the circle is the probability that -a*sqrt(3)/2 < x < a*sqrt(3)/2 and this is a*sqrt(3)/(2a) = sqrt(3)/2 Method (3) ---------- A point is chosen at random on the circumference of the circle and a tangent drawn at this point. Then a chord is drawn such that the angle between the chord and tangent is uniformly distributed between 0 and 180. If this angle exceeds 30 degrees and is less than 150 degrees then the chord is longer that the radius of the circle. So the probability that the chord exceeds the radius of the circle is 150 - 30 120 -------- = ----- = 2/3 180 180 And so we get 3 perfectly correct but different answers to the question. This example illustrates the very important fact that there may be equally valid but different ways of defining "randomness" and in some situations it is not at all clear which is the truly "random" method. - Doctor Anthony, The Math Forum http://mathforum.org/dr.math/ |
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