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"I just had an odd revelation in math today. I'm a seventh grader, and my teacher suggested I email a professor. We were doing some pretty basic math, comparing x to 3 and writing out how x could be greater, less than or equal to 3. But then it occurred to me; would that make a higher probability of x being less than 3? I mean, if we were comparing x to 0, there would be a 50% chance of getting a negative, and a 50% chance of being positive, correct? So, even though 3 in comparison to an infinite amount of negative and positive numbers is minuscule, it would tip the scales just a little, right?" ~ Ella from California
Good morning Ella,
This is a very interesting question! For the sake of exploring this idea, can we agree that we’re talking about just integers (In other words, our random pick of a number could be -7, or 8, but it can’t be 2.5 or 1/3)? You didn’t specify one way or the other, and limiting our choices to integers will make it simpler to reason it out.
I’d like to start by pointing out that doing a random selection from all integers is a physical impossibility in the real world. There are essentially three ways we could attempt it: mental, physical, and digital. All three methods are impossible to do.
Mental: Your brain is incapable of randomly selecting from an infinite (unbounded) set of integers. You’ll be far more likely to pick the number one thousand than (for example) any number with seven trillion digits.
Physical: Write integers on slips of paper and put them in a hat. Then draw one. You’ll be writing forever if you must have an infinite number of slips. You’ll never get around to drawing one!
Digital: As a computer programmer who develops games for this site, I often tell the computer to generate random numbers for me. It looks like this: number = rand(-10000, 10000), and it gives me a random integer between -10000 and +10000. But I can’t put infinity in there. Even if I could, it would require an infinite amount of storage to create infinitely large random numbers. (The same issue holds true for doing it mentally, by the way – your brain only has so much storage capacity!)
Okay, so having clarified that this is not a practical exercise, we have to treat it as purely theoretical. So let’s talk about theory. Mathematically, we define probability as follows:
Probability of event happening = (desired outcomes)/(possible outcomes).
For example, If I pull a card from a deck of cards, what’s the probability that it’s an Ace?
Probability of an Ace = 4/52, because there are 4 desired outcomes (four aces) out of 52 possible outcomes.
But here’s where we run into a problem. The definition of probability requires you to put actual numbers in. And infinity is not a number. I have hilarious conversations with my five-year-old son about this – someone told him about infinity, and he just can’t let go of the idea. "Daddy, infinity is the biggest number, but if you add one to it, you get something even bigger." Infinity can’t be a number, because you can always add one to any number, giving you an even bigger number, which would mean that infinity is actually not infinity, since there’s something even bigger.
So here’s where we’re at: we can’t do this practically, and we also can’t do it theoretically, using our definition of probability. So instead, we use a concept called a “limit” to produce our theoretical result. This may get a bit complicated for a seventh grader, so I'll forgive you if your eyes glaze over for the next couple paragraphs!
Let’s forget for a moment the idea of an infinite number of integers, and focus on integers in the range negative ten to positive ten. If we wanted the probability of picking a number less than 3, we’d have: Probability = 13/21, because there are 13 integers less than 3, and a total of 21 in all (ten negatives, ten positives, plus zero). What if the range was -100 to +100? Then Probability = 103/201. If the range was -1000 to +1000, we’d have 1003/2001.
Now let’s take this a step further and say that the integers range from -x to +x, where x is some integer we pick. The probability is (x + 3)/(2x + 1). Now we ask, “As x gets bigger and bigger, what does this fraction approach?” Mathematically, we write it as shown in the image below:
We'd read this as: "the limit as x approaches infinity of (x + 3) over (2x + 1)."
Evaluating limits like this is something my Pre-Calculus and Calculus students work on. Don’t worry, I’m not going to try to make you evaluate it – I’ll just send you here: Wolfram Limit Calculator. In the first textbox, type “inf” and in the second textbox, type (x + 3)/(2x + 1). Then click submit. The calculator will tell you that the limit is 1/2.
That’s probably not what you wanted to hear, right? You wanted me to tell you that the probability is just a tiny bit more than 1/2. And I sympathize with that – I’d like it to be more than 1/2 too! But remember that since infinity isn’t a number, we can’t plug it into our probability formula, so the probability doesn’t exist; only the limit of the probability exists. And that limit is 1/2.
Just for fun, if we could do arithmetic operations on infinity, I could solve it this way: “How many integers are there less than 3? An infinite number. How many integers are there three or greater? An infinite number. How many is that in all? Twice infinity. Therefore the probability is ∞/(2∞) = 1/2.” We can’t do arithmetic operations on infinity like that, because if we try, we eventually end up with some weird contradictions. But even so, it’s interesting that we end up with the same answer by reasoning it out that way!
PS - For clarification, "Professor Puzzler" is a pseudonym, and I'm not actually a professor. I'm a high school math teacher, tutor, and writer of competition math problems. So if your teacher needs you to contact an "actual professor," you should get a second opinion.