Paradoxes

by Tom Kando

I am fascinated by paradoxes. Sometimes, they can serve as amusing conversation pieces at a dinner party.

Here is a familiar one: “This sentence is false.” This is of course an exercise in circular logic: if the sentence is indeed false, then wouldn’t it be telling you the truth by telling you it was false? And if the sentence was true, then wouldn’t its declaration of falsity render it untrue?

But right now, I just want to select a few “numerical” paradoxes that I have had on my mind over the years. To mathematicians and statisticians (neither of which I am), this post will appear innumerate and dilettantistic. Nevertheless consider the following:

● My twin sisters were born two years after me. So when I was four years old, my sisters were 50% of my age. Today, because of our advancing years, their age is over 90% mine. Thus, as the three of us become older and older, my sisters’ age will eventually be 100% of mine - they will catch up with me and be exactly as old as I am! How can that be?

Well, calculus and integral mathematics tell us that it is only at the LIMIT where my sisters will be exactly as old as me, and this is in infinity.

● We owe many amusing paradoxes to Zeno, a Greek philosopher who lived 2,500 years ago. Many of you are probably familiar with Achilles and the tortoise: They race each other, but Achilles, who is twice as fast as the tortoise, starts one mile behind the tortoise. When Achilles reaches the line where the tortoise started, the tortoise meanwhile has advanced half a mile. When Achilles covers that half mile, the tortoise has moved up another quarter mile, etc. Thus, Achilles can never overtake the tortoise. But in reality, we all see that he does catch up, right?

Here, again, mathematicians introduce “infinities” to solve the paradox.

● This next one may not intrigue everyone, but I remember grappling with it in my statistics classes, and also when I went to Las Vegas: When you toss a coin, the chance of it coming out heads is 50%, and so is the chance of it coming out tails. Same with putting your money on black or red at the roulette table (forget the 0 and double 00 which gives the house a slight edge).

The probability of TWO consecutive identical outcomes (two heads in a row, or the roulette ball falling twice in a row on the same color) is 25%. The probability of THREE identical outcomes in a row is only 12.5%, and so forth: The probability of the roulette ball falling on the same color ten times in a row is less than 1 tenth of one percent.

So let’s say you watch a roulette table at play just as a spectator, and you see the ball falling on a black number three times in a row. Now you decide to play. Because the ball has just fallen on a black number three times in row, you figure it’s likely to be red’s turn finally, and you put your money on red. After all, the probability of FOUR consecutive black outcomes is only slightly over 6%, so you figure that if you put your money on red, your chances of doubling it are 94%!

And you are totally wrong. Your chances remain 50%, every time you play red or black. Even after one hundred consecutive black outcomes, the odds for black and red on the next play would remain the same - 50-50. (Actually, if the outcome were one hundred times black in a row, I would put my money on black, assuming that the roulette table was rigged).

Your error would be what is called the “gambler’s fallacy:” When statisticians tell us that the chance of a roulette ball falling on black four times in a row is only 6%, this probability refers to the entire series, NOT to an individual roll.

● And here is another amusing paradox: It’s obvious that there is an infinite amount of (whole) numbers, right? It’s also obvious that half of all the whole numbers are even (and the other half are odd).

But here is a third obvious fact: There is also an infinite amount of even numbers. So how can the totality of all numbers and the total amount of even numbers be the same (infinite), and yet one be only half as many as the other?

Again, mathematicians have to rescue us from our muddled brains: For one thing, “infinity” is not a number. For another, there are different sorts of infinities (see previous paradoxes).

So these are the four thoughts I chose to entertain you with today (or to give you a sleepless night?) leave comment here

© Tom Kando 2015