Crazy Bets and the Nature of Sure Things
Are you a betting person? I like to play craps in the casino. Not because I think I can win – the house has an edge in every game. But craps tables can be fun when everyone is rooting for the dice to hit a certain number, and then everyone shouts and high-fives one another. Lots of fun. Until a seven is rolled and everyone loses their money (except the non-socialites who are playing the “don’t”).
But, here’s a bet you can’t lose – if you get to make the offer.
If I were to offer you the following sum of money in dollars, would you take it? Here’s the sum:
1 + 2 + 3 + 4 + 5 + … and so on, forever.
So, that’s the infinite sum of one plus two plus three and so forth – in dollars. If I offered you that deal, would you take it? It’s a sure thing, right?
If you said yes, then guess how much money you’d have?
Answer: you would owe me eight cents.
Sounds crazy, right? Crazy, but true. And the math behind it is actually quite simple.
Here are the guys from Numberphile showing you a basic version of the proof:
What the video (and the mathematics) proves is that the infinite sum of 1 + 2 + 3 + 4 + … adds up to -0.08. That’s not a typo – the sum adds to negative 0.08 (actually, negative one-twelfth to be exact). It is so counterintuitive that a sum of all positive integers adds to negative one-twelfth. But it’s a fact.
Thus if someone took your “sure thing” bet, they would end up owing you about eight cents.
That’s the nature of sure things – they often aren’t.
It is so easy to allow our ego, greed, arrogance, naiveté, or desperation to believe certain things are true or will come to pass. And when they don’t, we often blame someone (or something) other than ourselves for our foolish thinking.
The next time you are absolutely certain about something – consider the foundation upon which your beliefs rest. If they’re solid, then you have nothing to worry about. Otherwise, a healthy dose of humble introspection can go a long way.
Morals of the story
- Don’t ever go to the casino to win – in the long run, you won’t. But if you want to have fun, go to a rowdy craps table.
- Don’t assume a “sure thing” is so sure. You might owe someone eight cents.
- A little knowledge of math can make you a lot of money.
What “sure things” have you encountered that turned out to be anything but?