Scientific American presents Math Dude by Quick & Dirty Tips. Scientific American and Quick & Dirty Tips are both Macmillan companies.

At the end of the last episode on how to calculate probabilities, I assigned you a little project about flipping 1, 2, 3, and finally 4 coins at once. Your first goal was to use a probability tree—or any other method you like—to figure out how many possible outcomes there are in each case. And your second goal was to figure out how many of these outcomes will give you 0, 1, 2, 3, or 4 heads. 

Why would you want to do that? I know this might sound kind of strange, but it’s because the answer you get is sort of surprising…and it’s very cool. Plus, it’s related to a famous and fascinating pattern you may have heard of called Pascal’s triangle. What exactly is that? And where does it come from? Stay tuned, because that’s precisely what we’re talking about today.

Tossing 1 and 2 Coins
The first thing we need to do on our quest to discover Pascal’s triangle is figure out how many possible outcomes there are when tossing 1 and 2 coins at the same time. Of course, when we toss a single coin there are exactly 2 possible outcomes—heads or tails—which we’ll abbreviate as “H” or “T.” How many of these outcomes give 0 heads? Well, 1 of them. And there’s also 1 outcome that gives 1 head.

Thinking about 1 coin is almost too easy, so let’s move on to 2 coins. As we saw when we first learned about probability and probability trees, there are 4 possible outcomes when tossing 2 coins. That makes perfect sense since each of the 2 possible outcomes for the first coin—H or T—has 2 possible outcomes for the second coin—again, H or T. So there must be twice as many outcomes for 2 coins as there are for 1 coin. And there are! HH, HT, TH, or TT.

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