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#### Pranav

##### Well-known member

- Nov 4, 2013

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(I am not sure if this is the right place to post it but since my solution involves some Calculus, I decided to post it here.)

(Also, I am not well versed with the correct words and terms to be used while doing geometrical probability problems, I am sorry if I write something wrong. )

Three points are selected on circumference of a circle. Find the probability that they lie on a semicircle.

$\angle POX=\theta$

$\angle QOX=\alpha$

$\angle ROX=\beta$

I select three arcs on the circle of length $Rd\theta$, $Rd\alpha$ and $Rd\beta$. The probability for P,Q and R to lie in these arc lengths is $\frac{d\theta}{2\pi}$, $\frac{d\alpha}{2\pi}$ and $\frac{d\beta}{2\pi}$ respectively. Hence,

$$dP=\frac{1}{8\pi^3}\,d\theta\,d\alpha\, d\beta$$

The limits for $\beta$ are from $-\pi+\alpha$ to $\pi+\theta$ but these limits only work when $\alpha$ lies from $\theta$ to $\pi+\theta$ and $\theta$ lies from 0 to $2\pi$. Solving the definite integral with these limits and multiplying the result by two gives the correct answer.

My question is whether there is way to identify the correct limits for $\alpha$ and $\beta$ in the first try. I spent quite a lot of time to figure out the correct limits. I initially used the limits $-\pi+\alpha$ to $\pi+\theta$ for $\beta$ and 0 to $2\pi$ for $\alpha$ which gives the incorrect answer. I tried checking different cases to see if my initial limits holds and then I arrived at the correct limits which took up a lot of time. Is there a nicer way to figure out the limits?

Any help is appreciated. Thanks!

(I have never formally studied triple integrals, the above integral was set up completely on my intuition so I am not sure if I have done it correctly.)

(Also, I am not well versed with the correct words and terms to be used while doing geometrical probability problems, I am sorry if I write something wrong. )

**Problem:**Three points are selected on circumference of a circle. Find the probability that they lie on a semicircle.

**Attempt:**$\angle POX=\theta$

$\angle QOX=\alpha$

$\angle ROX=\beta$

I select three arcs on the circle of length $Rd\theta$, $Rd\alpha$ and $Rd\beta$. The probability for P,Q and R to lie in these arc lengths is $\frac{d\theta}{2\pi}$, $\frac{d\alpha}{2\pi}$ and $\frac{d\beta}{2\pi}$ respectively. Hence,

$$dP=\frac{1}{8\pi^3}\,d\theta\,d\alpha\, d\beta$$

The limits for $\beta$ are from $-\pi+\alpha$ to $\pi+\theta$ but these limits only work when $\alpha$ lies from $\theta$ to $\pi+\theta$ and $\theta$ lies from 0 to $2\pi$. Solving the definite integral with these limits and multiplying the result by two gives the correct answer.

My question is whether there is way to identify the correct limits for $\alpha$ and $\beta$ in the first try. I spent quite a lot of time to figure out the correct limits. I initially used the limits $-\pi+\alpha$ to $\pi+\theta$ for $\beta$ and 0 to $2\pi$ for $\alpha$ which gives the incorrect answer. I tried checking different cases to see if my initial limits holds and then I arrived at the correct limits which took up a lot of time. Is there a nicer way to figure out the limits?

Any help is appreciated. Thanks!

(I have never formally studied triple integrals, the above integral was set up completely on my intuition so I am not sure if I have done it correctly.)

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