# Where Probability Meets Geometry, the infinite may be infinitesimal

Here’s a simple problem that I think could bring up a number of good discussions. I’m not sure what grade level this is — I’m not sure geometric probability is ever taught, except to kids practicing for math contests. But hopefully some of the HS teachers out there will be able to tell me where this fits.

ABCD is a square with side length 10. P is a point chosen at random inside ABCD. What is the probability that P lies on one of the diagonals of ABCD?

If you teach probabilistic geometry, would you give students a problem like this? Do you think they would know how to solve it? Do you think they would really understand the solution, or feel that it is some artificial thing mathematicians do like setting 0! = 1?

Oh but 0! = 1 is not artificial! It goes back to the gamma function. Here is a link to a (not entirely satisfying) discussion of it on Dr. Math — but I have it in my Integral Equations notes somewhere. . . . .

Nonetheless, it

feelsartificial when students are introduced to factorials in middle school. π I think the same thing happens when kids are introduced to the idea that probability can be 0 for something that is still “possible”. I think especially this case, it may seem odd to kids that the probability of choosing any one of the infinity of points on the diagonals is still zero, because as you said in the other post, the diagonals have no area.Thanks for your comments!

I think you’re right; 0! = 1 is very counter-intuitive. I always try not to say, “Mathematicians have agreed that….” but in this case, it’s not much different from, “There’s a good reason for this, but it involves calculus.” The link I included before has a much more intuitive explanation that I plan to use next year.

And back on topic, my statistics framework does call for us to teach geometric probability, and it’s also covered in our Algebra II book, though I don’t think we actually teach that section.

I didn’t use any problems like this when I taught this in the fall, but I think it’s a great idea. It’s kind of like “the exception that proves the rule.”

For 0!, I usually only introduce it if I’ve introduced the formulas for Permutations and Combinations (which I mostly prefer not to do), at which point I state that “Mathematicians have agreed…” in part because these formulas would break if it weren’t so. But you are right that that’s not all that satisfying. π

Is your statistics class an AP class? Or what grade level approximately is it intended for?

No, they’re not AP. (I would love to have an AP stats section at some point, though.) It’s a regular 2-semester course, offered by our school for 11th & 12th graders who have finished Algebra II but are likely too weak in (or just not interested in) trig/pre-cal. For the most part, my statistics classes are made up of students who rather dislike math; all of their pro-math classmates are in trig/pre-cal. Probability takes up about 25% of the course, closer to 50% if you count everything we do with the standard normal curve. I’m learning it along with them – I never had stats as a student, and I was drafted into it as the new hire this year. (No one else wanted to take on a new course.) But I’m glad, because I’m enjoying it. π

We don’t need the Gamma function to understand why 0! =1. It is similar, say, 4^0 = 1 in the sense that the only way to reasonably define 4^0 is to make the exponent rules continue to work (e.g. 4^n/4^m = 4^(m-n) ). For factorials, n! / n = (n-1)!, for n=2,3,4, …. so define 0! make it work for n=1.

oops that should 4^(n-m) of course…

Thanks, Ned, that is of course a simpler explanation of why we set the value as we do.