Illinois State University Mathematics Department

MAT 312: Probability and Statistics for Middle School Teachers

Dr. Roger Day (

Geometrical Probability

Geometrical probability refers to the use of geometrical representations and calculations to determine the probabilities of outcomes. The outcomes in question must be such that they can be represented directly or indirectly through one-, two, or three-dimensional geometrical shapes. The notes here are intended to provide illustrations and examples to help you conceptualize notions of geometrical probability and begin to answer your own probability questions using geometry.

Illustrative Example: A 1-Dimensional Case

If I give you a 5-inch piece of string and ask you to make one cut in the string at some random point along its length, what is the probability one of the resulting peices will be less than 1 inch long?

Here's an image of the string.

To address the probability question, we need to determine where on the string a "successful cut" will occur, where success means we will get a piece of string less than 1 inch after the cut.

The image above shows our regions of success: the last 1 inch on either end of the string. A cut in either of these 1-inch regions will result in a piece of string less than 1 inch long.

To determine the desired probability, we now compare the success region to the entire region: 2 inches to 5 inches. Thus our probability is 2/5 or 0.4.

Illustrative Example: A 2-Dimensional Case

Here's another probability question that lends itself to a geometrical interpretation:

Suppose we throw a dart at the board shown below. If we know the dart hits somewhere on the 18-inch square board, and that the dart has hit the board at random, what is the probability of a bull's eye? That is, what's the probability that a point chosen at random on the board is within the smallest circle?

We apply area formulas here to determine the region of success compared to the entire region. The bull's eye area, the success region, is just the area of a circle with radius 1 inch. The entire board is a square with side length 18 inches. Thus, our desired probability is (pi*1^2)/(18*18) or approximately 0.00969 if we use 3.14 as an approximation for pi.

Discrete or Continuous?

A question you ought to wonder about: What if the desired point--either the scissor cut or the dart, in these two illustrations--hits on a point on the border of a success region? This boils down to the distinction between continuous and discrete probability situations, a discussion we've yet to have this semester.

The short answer is that for our probability calculations, it doesn't matter, because the probability of hit within the circle, for the dart board problem, is the same as hitting on or within the circle. The width of the border line, although we can see it here and we can draw it with the thickness of a pencil point, is essentially 0.

Return to MAT 312 Homepage