Infinite spaces; infinitesimal probabilities
No one commented on my brief post about a simple but interesting geometric probability problem.
There are a lot of interesting things to discuss about probabilities when the sample space is infinite. Such as:
- What is the probability that if a student guesses randomly on a math problem whose answer could be any rational number, he will get the problem correct? (See my post on cheating on the Math Olympiad to see why that problem comes to mind.) 😦
- What is the probability that if an integer is chosen at random, it will be even?
- Divide a rectangle into two parts, one with twice the area of the other. Pick a point randomly within the rectangle. What is the probability that the point is inside the smaller part?
Problem 1. exemplifies is something that bothers “beginners” about probability — that something that is “possible” can have probability zero.
Problem 2. raises many interesting questions. We know that the cardinality of the set of even integers is equal to the cardinality of all the integers. (For more on this, check the links here.) Yet, intuitively we know that the answer must be 1/2. There’s a fascinating discussion of this problem and more complicated versions at The Math Forum‘s Ask Dr. Math archive. In particular, I was surprised to read the following: “If you’re talking about the entire infinite set of integers, there is no way to do this [choose an integer at random] without some sort of a distribution function over the integers, and there is no such function that gives an “equal probability” for all integers.” (emphasis mine) The problem is solvable, of course, by using limits. And the answer, as we expect it must be, is indeed 1/2.
Problem 3 is a “basic” geometric probability problem. It was raised by another Ask Dr. Math reader (follow the link in the question) who argued that since the cardinality of each of the spaces is the same, the probability of landing in each must be the same. Doctor Tom replied, in part that “the cardinality of the set does not determine its measure” and that it is “measure” and not “cardinality” that is important in determining probabilities.
Then we come back to my original question:
- 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?
What I think is cool about this question is that the number of points on the diagonals is infinite, but
it it’s size/area/measure (corrected in response to Pseudonym’s comment) is infinitesimal in comparison to the size of the area of the square. I suppose in that way it’s related to the problem below about choosing a rational number at random out of the set of all real numbers.
Some other interesting and related questions from Ask Dr. Math:
- Imagine a line extending infinitely in both directions. A line segment of length 10m has endpoints at point A and Point B, both of which are on the line. What is the probability that a randomly chosen point on the line is on line segment AB?
- What is the probability of choosing a random rational number from the set of reals?
- Take a line segment. Divide it into three segments such that one is half the length of the line and the other two are a quarter of the line segment. Choose a point at random along this line segment. What is that probability that this point lands in the 1/2 segment?
And my favorite:
- Three randomly drawn lines intersect so as to form a triangle on an infinite plane. What is the probability that a randomly selected point will fall inside that triangle?
Please post your comments on any of these problems that pique your interest or raise issues that you’d like to discuss further!