What is the probability that a random chord is the diameter of a circle?

If we choose a chord at random from a circle (let us use the unit circle to simplify the question), what is the probability that this chord would be a diameter of this circle?

My thinking is that the length of a random chord is a continuous random variable, and as such we could only determine interval probabilities for it (for example, what is the probability that a random chord is longer than $\sqrt{3}$ in the unit circle, a rephrasing of Bertrand's paradox). The probability of the length being fixed (or equal to the diameter of the circle), however, should be zero. As any circle has an infinite number of chords and an infinite number of diameters, I am not totally sure whether this creates an indeterminacy in the calculation of the probability.

#Geometry

Easy Math Editor

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Comments

Suppose we fix one end of the chord at $(x,y) = (1,0)$ , so that only the other end of the chord is variable. Then the probability of the chord being a diameter is obviously zero. It's the fixing of the one end that makes it simpler for me.

Steven Chase - 2 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$