# Flat Probabilities on a Sphere

### Problem

### Answer

The probabilities in the problem are defined by the central angle $\alpha$ between the two selected points. The probability of winning is defined by the function

$\displaystyle p(\alpha)=\sin\left(\frac{\alpha}{2}\right)+\frac{1}{2}\left(\cos\left(\frac{\alpha}{2}\right)-1\right).$

The function attends its maximum for $\displaystyle\alpha_o=2\arctan(2)\approx 126.9^{\circ}.$ In addition, $\displaystyle p(\alpha_o)=\frac{1}{\varphi}=\frac{\sqrt{5}-1}{2}.$

### Solution

Suppose you choose two points with central angle $\alpha.$ By symmetry we may assume they have coordinates $(0,b,c)$ and $(0,b-c).$ The diagram below shows a 2d cross section.

You lose your bet if your friend selects a point in one of three caps: above the plane $z=c,$ below $z=-c$ or to the right of $y=b.$

Thus, we need to compute the areas of these three spherical caps. We do that using a theorem discovered by Archimedes:

If two parallel planes $h$ units apart slice a sphere of radius $r,$ the surface area between the planes is $2\pi r h.$

Using these calculations, we conclude that the probability that you win the bet is

$\displaystyle p(\alpha)=\sin\left(\frac{\alpha}{2}\right)+\frac{1}{2}\left(\cos\left(\frac{\alpha}{2}\right)-1\right).$

The function attends its maximum for $\displaystyle\alpha_o=2\arctan(2)\approx 126.9^{\circ}.$ In addition, $\displaystyle p(\alpha_o)=\frac{1}{\varphi}=\frac{\sqrt{5}-1}{2}.$

### Acknowledgment

This problem is an adaptation of Problem 305 (a Jungle Gym problem, *Math Horizons* n 32, 2014). The problem and the solution are by Randy Schwartz of Schoolcraft College (MN).

Steve Phelps created beautiful interactive illustrations for the problem. The latest can be accessed at the GeoGebra site; the same holds for the earlier version.

### Golden Ratio

- Golden Ratio in Geometry
- Golden Ratio in Regular Pentagon
- Golden Ratio in an Irregular Pentagon
- Golden Ratio in a Irregular Pentagon II
- Inflection Points of Fourth Degree Polynomials
- Wythoff's Nim
- Inscribing a regular pentagon in a circle - and proving it
- Cosine of 36 degrees
- Continued Fractions
- Golden Window
- Golden Ratio and the Egyptian Triangle
- Golden Ratio by Compass Only
- Golden Ratio with a Rusty Compass
- From Equilateral Triangle and Square to Golden Ratio
- Golden Ratio and Midpoints
- Golden Section in Two Equilateral Triangles
- Golden Section in Two Equilateral Triangles, II
- Golden Ratio is Irrational
- Triangles with Sides in Geometric Progression
- Golden Ratio in Hexagon
- Golden Ratio in Equilateral Triangles
- Golden Ratio in Square
- Golden Ratio via van Obel's Theorem
- Golden Ratio in Circle - in Droves
- From 3 to Golden Ratio in Semicircle
- Another Golden Ratio in Semicircle
- Golden Ratio in Two Squares
- Golden Ratio in Two Equilateral Triangles
- Golden Ratio As a Mathematical Morsel
- Golden Ratio in Inscribed Equilateral Triangles
- Golden Ratio in a Rhombus
- Golden Ratio in Five Steps
- Between a Cross and a Square
- Four Golden Circles
- Golden Ratio in Mixtilinear Circles
- Golden Ratio With Two Equal Circles And a Line
- Golden Ratio in a Chain of Polygons, So to Speak
- Golden Ratio With Two Unequal Circles And a Line
- Golden Ratio In a 3x3 Square
- Golden Ratio In a 3x3 Square II
- Golden Ratio In Three Tangent Circles
- Golden Ratio In Right Isosceles Triangle
- Golden Ratio Poster
- Golden Ratio Next to the Poster
- Golden Ratio In Rectangles
- Golden Ratio In a 2x2 Square: Without And Within
- Golden Ratio With Two Unequal Circles And a Line II
- Golden Ratio in Equilateral and Right Isosceles Triangles
- Golden Ratio in a Butterfly Astride an Equilateral Triangle
- The Golden Pentacross
- 5-Step Construction of the Golden Ratio, One of Many
- Golden Ratio in 5-gon and 6-gon
- Golden Ratio in an Isosceles Trapezoid with a 60 degrees Angle
- Golden Ratio in Pentagon And Two Squares
- Golden Ratio in Pentagon And Three Triangles
- Golden Ratio in a Mutually Beneficial Relationship
- Star, Six Pentagons and Golden Ratio
- Rotating Square in Search of the Golden Ratio
- Cultivating Regular Pentagons
- Golden Ratio in an Isosceles Trapezoid with a 60 degrees Angle II
- More of Gloden Ratio in Equilateral Triangles
- Golden Ratio in Three Regular Pentagons
- Golden Ratio in Three Regular Pentagons II
- Golden Ratio in Wu Xing
- Golden Ratio In Three Circles And Common Secant
- Flat Probabilities on a Sphere
- Golden Ratio in Square And Circles
- Golden Ratio in Square
- Golden Ratio in Two Squares, Or, Perhaps in Three
- Golden Ratio in Isosceles Triangle
- Golden Ratio in Circles
- Golden Ratio in Isosceles Triangle II
- Golden Ratio in Yin-Yang

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