I just came across this paradox in an old book of mathematical recreations and thought it was cute so I’d share it here:

Here are two possible solutions to pick from:

Since we are now in the era of precision cosmology, an uncertainty of a factor of 400 is not acceptable so which answer is correct? Or are they both wrong?

### 23 Responses to “A Paradox in Probability”

1. The second (area-based) argument seems correct to me; the first must be wrong because there is “more sphere” assigned to the intermediate angles of the great circle than to the very small or very large (for the same reason there is less area per latitude angle near the polar caps than near the equator).

2. Surely (a) is completely wrong? It seems to consider a 2D problem, so the result is correct for the question “if two points are chosen at random on the circumference of a CIRCLE….”? Constraining the great circle to pass through the two points on a sphere doesn’t address the question.

• telescoper Says:

But a (unique) great circle can be drawn through any two points on a sphere….

• Exactly…it is unique, and the subset of great circles which pass within 10′ of the second point is a fraction of about 1/400 of the full set of great circlesthrough the first point…

• Not really. It is not the same great circle going through any two points on a sphere.
Once you reduce the problem to any two points located on a unique great circle, the problem is indeed back to a 2D case…

• 0.000926 is what you get in the “two random points on a circle” case (120 arc minutes/360degrees). If the sphere case gives the same result, it means the sphere and the circle are degenerate and the whole geometry of the universe collapses!

3. Anton Garrett Says:

Depends what you mean by random. I’d say that (b) corresponds to the intuitively obvious meaning.

4. sorry… typo… 20 arc minutes/360 degrees

5. Jasjeet Singh Bagla Says:

This is identical to the calculation of two people in a group with birthday on the same day, and two people in a group with a birthday on a given day. No?

6. justcuriousenough Says:

Interesting! I’ll also go for the second, agreeing with Chris C.
Which book is that?

7. Both seem quite wrong to me, in different ways, thus the difference.

8. Method a seems to be equivalent to sampling phi and theta of the spherical coordinate system from uniform distributions, while method b is uniform sampling on the surface of sphere. Both are perfectly fine ways to pick “random” points, but method b is probably what most people would mean by picking random points if nothing else is said about the distribution.

• If you want a random point on a sphere should you make a uniform sample according to the measure $\sin(\theta)d\theta\dphi$ rather than just $d\thetad\phi$.

• Yes, a is sampling from d\theta\phi, while method b is from \sin(\theta)d\theta\d\phi.

But the question says just random, not uniformly random, so it’s hard to argue that any distribution is wrong.

9. How about a numerical simulation? I’m sure someone must have a tool that could set it up in a few minutes?

10. Le paradoxe de Bertrand, in 3D… https://www.cut-the-knot.org/bertrand.shtml

11. Anton Garrett Says:

I have Northrop’s book, and recommend also “Challenging Mathematical Problems with Elementary Solutions” by A.M. Yaglom and I.M. Yaglom, and Frederick Mosteller’s “Fifty Challenging Problems in Probability”.

The ancient Greeks knew what was the common volume of three indefinitely long cylinders of unit radius with axes along Ox, Oy and Oz, ie calculus is not needed. (It *is* needed for two intersecting cylinders.) And in Euclid’s Elements is the proof that if (2^n – 1) is prime then itself multiplied by 2^(n-1) is perfect. (It took another 2000 years for Euler to prove that all even perfect numbers are of this form, and we *still* don’t know if any odd perfects exist.) Also in Euclid is the ingenious proof that there is an infinite number of primes: a method is given how you can always construct one more.

12. […] little paradox in probability that I posted earlier in the week seemed to go down quite well so I thought I’d try a […]