Pick a number
Odd coincidences happen all the time. The other day I was on my way to Tulsa and had stopped at the so-called “World’s Largest McDonalds” that is on I-44 down around Vinita. I was in the mood to put another coat of cholesterol on my arteries and my family wasn’t along to perform an intervention. As a consequence I was in line to get a Quarter-Pounder combo with Fries, when I looked up an saw Bitty Bubba in the next line over.
Bitty Bubba, as you may recall, is one of my old friend Bubba’s nephews. He is the great hope of the family, so, as a consequence, the sum total of the family’s desires and fears are resting on his frail shoulders.
He was alone there too, so we sat down to eat together and began to talk. As it turns out, he’s taking a class in elementary probability and had learned a game he wanted to show me,
“Guess a number between one and three,” he directed.
“Inclusive or exclusive?” I asked. Whatever other interests I might have, I will be a mathematician until I die, so I make sure I know the rules before I get involved in a game. Bitty Bubba was a little confused by my question, however.
“What?” he asked.
“Inclusive means you include the endpoints—one and three in this case—and exclusive means you exclude them.”
“Uh...inclusive I guess,” he said.
“Two?” I replied.
“No,” he laughed. “I was thinking of 2.5.”
He had me. I’d been careful to ask about the endpoints, but I’d neglected to ask what he’d meant when he said “number,” assuming that he meant integer. You know what they say, if you assume you make an ass of you and me.
“That’s pretty funny,” I said. I appreciated his humor, but I didn’t want to waste a teaching opportunity. “Suppose, though, we’d stuck to the integers. What would’ve been my probability of getting a correct answer then?”
“Well,” he said, drawling it out to give himself time to think, “it would be one-third because there were three numbers to choose from—one, two, and three—but only one right number.”
“You are correct,” I said. I wasn’t surprised. In my experience as a math teacher, most students just know this. “Now, I’ve got a harder question to ask. You increase the numbers to those with a decimal representation. What is the probability of my choosing correctly then?”
His eyebrows nit for a moment like he was thinking very intently.
"I want to say zero," he finally answered, "but I don't know why."
"You are correct," I answered him. "The probability is zero. You can think about it like this. If you had ten numbers, the probabilty would be one tenth and if you had 100 numbers, it would be one one-thousandth, but there are infinitely many decimal numbers between one and three. When you say there are infinitely many numbers, that means whatever number you name there are more decimal numbers than that. That means the probability is smaller than one tenth or one one-thousandth or one one-millionth and so on. The probability is smaller than any number you can think of, so it's zero."
Bitty Bubba got that look in his eyes that so many of my students get when they are really fascinated with what I am saying. Oddly enough, it reminds a lot of people of a deer caught in headlights.
"Well, that's really interesting, but..."
"Do you know what is more interesting?" I asked. I hated to waste this teaching opportunity.
"But I really..."
"The probability of choosing a decimal number at random and getting a repeating decimal is zero as well," I said.
"Repeating decimals?" Bitty Bubba asked.
"Yes, repeating decimals are those like 1.222222... where the 2s go on forever or 2.31234343434...where the 34s repeat forever. There are infinitely many repeating decimals, there are even more that don't repeat. In fact, the number that are repeating is somehow insignificant to the the number that doesn't."
I was waxing eloquent now and look on Bitty Bubba's face was more mesmerized than ever.
"Indeed, we need new concepts to make sense of this. Instead of cardinality that we use when we count the members of a set, we need to use the concept of measure..."
"Oh!" Bitty Bubba cried out. "My bus is leaving."
With that he rushed from the dining area.
I thought this was strange because I hadn't even seen any buses. I thought it was even stranger when I saw someone that looked a lot like Bitty Bubba riding down the interstate below me on a motor cycle.