# Geometric Probability

Random events that take place in continuous sample space may invoke geometric imagery for at least two reasons: due to the nature of the problem or due the the nature of the solution.

Some problems, like Buffon's needle, Birds On a Wire, Bertrand's Paradox, or the problem of the Stick Broken Into Three Pieces do, by their nature, arise in a geometric setting. The latter also admits multiple reformulations which require comparison of the areas of geometric figures. In general, we may think of *geometric probabilities* as non-negative quantities (not exceeding 1) being assigned to subregions of a given domain subject to certain rules. If function μ is an expression of this assignment defined on a domain D, then, for example, we require

0 ≤ μ(A) ≤ 1, A ⊂ D and

μ(D) = 1

The function μ is usually not defined for all A ⊂ D. Those subsets of D for which μ is defined are the *random events* that form a particular sample spaces. Very often μ is defined by means of the ratio of areas so that, if σ(A) is defined as the "area" of set A, then one may set

### Problem 1

Two friends who take metro to their jobs from the same station arrive to the station uniformly randomly between 7 and 7:20 in the morning. They are willing to wait for one another for 5 minutes, after which they take a train whether together or alone. What is the probability of their meeting at the station?

In a Cartesian system of coordinates (s, t), a square of side 20 (minutes) represents all the possibilities of the morning arrivals of the two friends at the metro station.

The gray area A is bounded by two straight lines,

[400 - (15×15/2 + 15×15/2)] / 400 = 175/400 = 7/16.

### Problem 2

([Sveshnikov, problem 3.12].)

Three points A, B, C are placed at random on a circle of radius 1. What is the probability for ΔABC to be acute?.

Fix point C. The positions of points A and B are then defined by arcs α and β extending from C in two directions. A priori we know that

Region D is the intersection of three half-planes:

Now observe, that unless the random triangle is acute it can be thought of as obtuse since the probability of two of the three points A, B, C forming a diameter is 0. (For BC to be a diameter, one should have

Three points A, B, C are placed at random on a circle of radius 1. What is the probability that all three lie in a semicircle?

### References

- E. J. Barbeau, Murray S. Klamkin, W. O. J. Moser, Five Hundred Mathematical Challenges by (MAA, 1995, problem 244.)
- D. A. Klain, G.-C. Rota
*Introduction to Geometric Probability*, Cambridge University Press, 1997 - A. A. Sveshnikov,
*Problems in Probability Theory, Mathematical Statistics and Theory of Random Functions*, Dover, 1978 - A. M. Yaglom, I. M. Yaglom,
*Challenging Mathematical Problems With Elementary Solutions*, Dover, 1987

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