Find out how to solve last week’s brain teaser about gambling, and learn what probability trees are and how to make them.
At the end of the episode called What Is Probability?, we imagined that a friend challenged you to an extremely simple game of chance. The goal of the game was for you to guess whether each toss of a coin would come up heads or tails. Since you knew that the probability of the coin landing heads and tails is the same, you decided to guess heads every time. But then, after heads came up four times in a row, you started to question your plan—after all, the coin was due to come up tails at that point, right?
Did you decide to switch to tails? Or did you stick with heads? And, most importantly, did the decision you made actually matter? Or were your chances of winning the same either way? Stay tuned because these questions and more are exactly what we’ll be talking about today.
The Gambler’s Fallacy
As we learned in What Is Probability?, probabilities are decimal numbers between 0 and 1 that tell you how likely it is that something will happen. The higher the probability, the more likely it is that whatever we’re talking about will happen. A probability of 0 means there’s literally no chance of it happening, and a probability of 1 means it’s certain to happen. For example, since a coin you toss in the air is equally likely to land heads or tails, the probability of either outcome is 1/2. And, believe it or not, that’s all you need to know to decide whether or not you should switch your guess in your friend’s game. So, what should you do?
The short answer is that it really doesn’t matter what you decided to do because the probability of the next toss coming up heads—or tails—is always 1/2. Which means that even after 4 consecutive heads, you still had a 50/50 chance of guessing right…no matter what choice you made. After all, how could the coin know that it’s supposed to come up tails this time just because it had landed heads 4, 40, 400, or however many times in a row? Obviously it can’t because it doesn’t know anything at all about its amazing streak! Nevertheless, it’s easy to let yourself believe that it can because it really does feel like the coin is due. And that feeling has cost a lot of people a lot of money. Which is precisely why this little puzzle is sometimes known as the “Gambler’s Fallacy”—don’t be caught in its trap!
What Makes an Outcome Unlikely?
Here’s something kind of counterintuitive: Although it seems unusual to have a coin come up heads 10 times in a row, that outcome is really no more unusual than any other outcome. In other words, tossing something like heads then tails then heads then tails for 10 flips—or any other sequence, for that matter—is just as unlikely as anything else. Why is it then that we perceive 10 straight heads to be especially unusual? Because it doesn’t look “random” like most other possible outcomes. And that non-random appearance preconditions us to think of it as an unusual result before we ever start flipping coins.
What Are Independent Events?
The key thing to realize with all of this is that each toss of a coin is a completely independent event. Which, as I alluded to earlier, means that each toss has no memory of the previous one and therefore cannot influence future tosses. How do we know that? Is it possible that coins have some mechanism for remembering the past and making decisions about the future? Perhaps those heads of former presidents on coins contain tiny brains that do the thinking?
Sadly, they don’t. The beauty of math and science is that we don’t just have to guess about the answer to a question like this. We can actually test it out by looking to see if there’s any connection between the outcome of one coin toss and the next. And if you actually do that kind of test you’ll find that there is no obvious connection. So each coin toss is indeed a mathematically independent event.
Probabilities for Tossing 2 Coins
Once we accept this about coin tosses, we can start to calculate the probabilities of all sorts of things. For example, let’s think about all the possible results we can get when we toss a coin twice. Namely:
toss 1 and toss 2 are both heads (which we’ll call HH)
toss 1 and toss 2 are both tails (TT)
toss 1 is heads and toss 2 is tails (HT)
toss 1 is tails and toss 2 is heads (TH)
So there are a total of 4 possible outcomes. If we assume that each toss is independent of the others (so that any one of these 4 outcomes is just as likely as the others), we can conclude that the probability of tossing 2 heads is 1/4 (since there is 1 outcome—HH—that gives 2 heads out of the 4 total possibilities), the probability of tossing 2 tails is 1/4 (again, since there is 1 outcome—TT—that gives 2 tails out of the 4 total possibilities), and the probability of tossing 1 head and 1 tail in either order is 1/4 + 1/4 = 1/2 (which we get by adding together both 1/4 probabilities from HT and TH).
While it’s not too tough to keep all of this straight in your head when there are only 2 coins, things start getting more difficult when the number of coins increases. One way to keep yourself from going crazy is to draw what’s called a probability tree. To get us ready to deal with those more complicated situations, let’s learn to sketch out a probability tree for our 2 coin scenario.
How to Draw a Probability Tree
To start, draw a nice big dot on a sheet of paper. That dot is the starting point from which we’re going to branch out and show all the possible results of flipping a coin twice. In our case, we start by flipping the coin to get either heads or tails. To represent these 2 possibilities, let’s draw 2 arrows extending to the right of our dot. Label the tip of the top arrow “heads” and write the probability of it happening—that’s 1/2, right?—along the arrow, and do the same for the bottom arrow representing the outcome of “tails.”
Now we’re ready to flip the coin a second time. What can happen this time? Well, for either outcome of the first flip, we can get “heads” or “tails.” We show this in our probability tree by drawing 2 arrows for “heads” and “tails” extending from the tip of the “heads” arrow of the first toss, and two more “heads” and “tails” arrows extending from the tip of the “tails” arrow of the first toss. Once you label these new arrows with the outcomes they represent and their probabilities, you’re done!
That’s all the math we have time for today. But never fear because next week, we’ll learn how to use these wonderful probability trees to make your probability calculating life a whole lot easier.
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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!