Are you a fan of brain teasers? If so, you’ll love today’s probability puzzler. And if not, hopefully you will be a fan after today. Regardless, have fun!
In past episodes, we've talked a lot about my all-time favorite recreational math writer, Martin Gardner, and the truly awesome body of work he produced for his monthly “Mathematical Games” column in Scientific American and the countless books of math puzzles that followed. If you’ve never picked up any of Mr. Gardner’s books, you really should … they’re pretty much guaranteed to bring you hours of fun.
As I like to do every so often, today I’m going to pull one of his wonderful books from my bookshelf and share a puzzle with you—and hopefully inspire you to want to think through even more of them on your own. The puzzle we’re going to talk about today comes from a book with the very appropriate title Entertaining Mathematical Puzzles. This particular entertaining puzzle is called, “The Three Pennies.” And, as you might guess, it’s all about the probabilities of tossing three coins.
The Three Pennies
Mr. Gardner presents this puzzle as a dialogue between a couple of guys called Joe and Jim. Joe kicks off the conversation with a proposition for Jim. He says, “I’m going to toss three pennies in the air. If they all fall heads, I’ll give you a dime. If they all fall tails, I’ll give you a dime. But if they fall any other way, you have to give me a nickel."
Should Jim take Joe up on his offer?
The question is: Should Jim take Joe up on his offer? Is it a good bet? I suppose if they’re only going to play one round of this game, it doesn’t really matter. Because the most Jim could lose is 5 cents. But what if they’re going to sit down for a long night of penny tossing and play 100 games? Or 1,000 games? Or 1,000,000 games? Then, we could be talking about big money.
So, what should Jim do?
The Wrong Answer
Jim takes the bet. And, as Mr. Gardner tells us in his book, that was the wrong—or at least the unwise—thing for him to do. Because it’s a bad bet.
Here’s the rationale that we are told Jim used to reach his ill-advised decision to play the game. First, Jim reasons that after any tossing of three coins, two of them must be the same. In other words, there must be either two heads or two tails. This is, of course, true. But he then goes on to reason that since the third coin has a 50/50 shot at being either heads or tails, then there’s a 50/50 chance that the third coin will match the two others and give him the win. Since Jim has been offered a dime if he wins and he only has to pay Joe a nickel if he loses, he figures it’s a good bet.
This seems like a reasonable thought process, right? But there’s a flaw in it somewhere … a very serious flaw. Can you see where Jim’s thinking has gone awry? Take a minute to think about it, and then continue on when you’re ready for the answer.