# Probability That A Randomly Selected Chord Exceeds The Length Of The Radius Of Circle

**Situation**

If a chord is selected at random on a fixed circle what is the probability that its length exceeds the radius of the circle?

- Assume that the distance of the chord from the center of the circle is uniformly distributed.
A. 0.5 C. 0.866 B. 0.667 D. 0.75 - Assume that the midpoint of the chord is evenly distributed over the circle.
A. 0.5 C. 0.866 B. 0.667 D. 0.75 - Assume that the end points of the chord are uniformly distributed over the circumference of the circle.
A. 0.5 C. 0.866 B. 0.667 D. 0.75

**Answer Key**

Part 1: [ C ]

Part 2: [ D ]

Part 3: [ B ]

Part 2: [ D ]

Part 3: [ B ]

**Solution**

**Part 1**

$d^2 + \left( \dfrac{r}{2} \right)^2 = r^2$

$d = \sqrt{r^2 - \dfrac{r^2}{4}}$

$d = \dfrac{\sqrt{3}r}{2}$

$P = \dfrac{d}{r} = \dfrac{\sqrt{3}r/2}{r}$

$P = \dfrac{\sqrt{3}}{2} \approx 0.866$ ← [ C ] *answer for part 1*

**Part 2**

$P = \dfrac{\pi d^2}{\pi r^2} = \dfrac{(\sqrt{3}r/2)^2}{r^2}$

$P = \dfrac{3}{4} = 0.75$ ← [ D ] *answer for part 2*

**Part 3**

If one end of the chord is at *A*, the other end must be on arc *BDC*.

$P = \dfrac{4}{6}$

$P = 0.667$ ← [ B ] *answer for part 3*

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