GATE Overflow Test Series | Probability | Test 1 | Question: 12

Question

If three points are randomly chosen along the circumference of a unit circle, dividing the circle into three arcs, the expected length of the arc that contains the point $(1, 0)$ will be ____

• A. $\frac{\pi}{3}$
• B. $\frac{2\pi}{3}$
• C. $\pi$
• D. $\frac{\pi}{2}$

Comments

For further reading –
 https://math.stackexchange.com/questions/272927/expected-length-of-arc-in-a-randomly-divided-circle
https://math.stackexchange.com/questions/13959/if-a-1-meter-rope-is-cut-at-two-uniformly-randomly-chosen-points-what-is-the-av

Answer 1

Lets break this circle and make it a straight line of length $2\pi$ at the point $(1,0).$  Now, $(1,0)$ is present at both ends and if we choose $3$ random points along this line, we are dividing this line into $4$ parts with all their lengths equally likely to be $\frac{2\pi}{4}.$ Since, two of the parts are favorable, our required expectation will be $\dfrac{1}{4} \times 2 \times 2\pi = \pi.$

Comments

These type of question will be easy on converting circle into line segment

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I strongly agree, but we should convince ourselves that how things are working without the circle-breaking method too.

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why we are multiplying by 2?

Answer 2

.