50/50 - or - How Three Coins Broke a Brain

My brain broke today. What did it? Admitting, to myself, that I've been wrong about something for more than ten years.



A very very long time ago, I was on a website that offered clever math puzzles. I was a nerd, that's what I did for fun. One went like this:

I have three coins. One is normal; one has two heads; the last has two tails.
You pull a coin at random and flip it. It lands heads-up. What are the odds that the other side is tails?

50/50, I said, and have always said. We know it's not the all-tails coin, so it's either the normal coin or the all-heads coin. One or the other, even odds of each.

The "official" answer disagreed. "one in three" it said. It claimed that it depended on the number of sides available or some such bullshit, a position I found and still find utterly laughable. The sides are  not mobile; it's one coin or the other. if it's the heads coin, both sides are always heads. The tails side can't sneak in there. Likewise for the normal coin. Completely absurd. I have held this position for years now: I am right, website was wrong.

And I never let it go. My mother was all too happy to accept the website's "available surfaces" theory. I couldn't accept it, and I couldn't stand being known as "wrong" when I knew I was right, and my mother was very clear about my wrongness. Neither of us ever relented from our position.

So last week, I got a little excited, a little ainxous, when Car Talk's puzzler echoed the same puzzle from across the mists of time. They framed it as a gamble: the man running the game bets even money that the other side is the same as the shown side. Do you take the bet?

I knew, I just knew, that they would say you shouldn't. That's how the whole thing was framed: you can't do better than 50/50 in any case, so it's not a great bet. So i was not surprised this week when Click and Clack confirmed my suspicion: bad bet, one-in-three odds. They even mentioned something like the mobile-faces theory again, though without detail as if they didn't buy it either. Why would they? these are intelligent people. But they bought the solution.

So I ran the scenario in my head, imagining a series of games: you're the gambler. You bet every time that the showing face matches the hidden face. And you win...

Every time a false coin is chosen. Which is two thirds of the time.

Hear that? That's what brains sound like when a crack forms.

Overall, it doesn't actually matter which face comes up. If the mark picks a bad coin, they lose. Waiting until a face shows makes it seem like the odds are better, but the overall odds only apply to the three coins together before any are drawn. If you say, "what are the odds you picked the genuine coin out of the original three", it's clearly one in three eliminating one of them doesn't change the coin it actually is. But somehow, the solution is never phrased in such clear terms.

For my own benefit, because I can look at my solution now and still question it, I ran a practical trial of my principled answer and found it to be correct. I got three coins and marked one as heads-heads and one as tails-tails and jumbled them and did the experiment. It wasn't perfectly proportioned but it proved the idea: the normal coin was picked about a third of the time, and on only half of its picks it was facing heads-up.

So to answer the original problem from years ago:

Overall there is a 1/3 chance of any one coin. If you eliminate the tails-tails coin, you have 50% chance of picking the normal coin, but only 50% of those draws count toward the original problem. The sides are not mobile; the fact that the two-headed coin has two heads just means that it's more likely to show a head at a random draw. In my trials, eliminating all draws that showed tails-up, it beat the normal coin by about two-to-one.

This practical trial would have resolved the problem years ago but I didn't do it at the time. Perhaps, subconsciously, I knew I would be proven wrong.

It's hard to admit you're wrong, especially when you know you're right. And sometimes you're never more certain of being right than when you are in fact quite wrong.