Martin Gardner’s 3 Penny Puzzle
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.
The Right Answer
As has already been said, Jim should have refused this bet. Because, unfortunately, he doesn’t understand how to calculate probabilities very well. In particular, he lacks an understanding of the distinction between permutations and combinations that we talked about a few weeks ago.
Jim should have refused this bet.
To see where Jim went wrong, let’s think about the number of possible permutations in the heads-tails results of tossing three coins. Imagine you toss the three coins one at a time. The first one could be either heads or tails. So, there are two possible outcomes. You then toss the second coin. There are two possible outcomes for that coin for each possible outcome of the first coin. So there are a total of 2×2=4 possible outcomes for two coins. Similarly, there are two possible outcomes for the third coin, which makes a total of 2x2x2=8 different possible heads-tails permutations when you toss three coins. As you can check, the possible permutations are:
- HHH
- HHT
- HTH
- HTT
- THH
- THT
- TTH
- TTT
Jim only wins if the three coins come up HHH or TTT (all the same), which will happen on average two out of every eight times the coins are flipped (this is the number of outcomes he needs compared to the total number of possible outcomes). So, over the long haul, Jim should only expect to win 1/4 of the time, which is precisely why Joe’s offer is not a good one for him.
Permutations Versus Combinations
So, where did Jim go wrong? It all comes down to the fact that he confused permutations and combinations. As we learned a few weeks ago, when it comes to combinations, the ordering of things doesn’t matter. So, if you think about it (and look at the list of possible outcomes of the coin toss), you’ll see that there are only four possible combinations when tossing three coins. Namely, all 3 heads, all 3 tails, 2 tails and 1 heads, or 2 heads and 1 tails.
Jim’s mistake was to think that these four combinations were all that matters when it comes to calculating the overall probability of him winning a game. But they’re not. It’s the number of permutations that enters into the probability equation … it’s the number of permutations that matters. As Jim has found, probabilities can be tricky to calculate. And it’s the subtle little things like this that often are the difference between winning and losing, which is all the more reason to understand the difference between combinations and permutations!
Wrap Up
OK, that’s all the math puzzling we have time for today.
For more fun with math, please check out my book, The Math Dude’s Quick and Dirty Guide to Algebra. And remember to become a fan of The Math Dude on Facebook, where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
Pennies image courtesy of Shutterstock.
You May Also Like…
About the Author
Jason Marshall is the author of The Math Dude’s Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.
