How to Calculate Probabilities, Part 2

Learn how to solve last week’s brain teaser about using probability trees to decide if and when you should go to the beach. Then use a probability tree to discover a famous and fascinating pattern in the probabilities of flipping coins.

Jason Marshall, PhD
Episode #122


by Jason Marshall

After the last few episodes, you should be feeling like you’re becoming something of a probability tree expert. Two weeks ago we learned how to draw probability trees, and last week we started to learn how to use probability trees to calculate probabilities. At the end of the last episode, I left you to think about a little brain teaser puzzle about using probability trees to help you decide if and when to go to the beach.

Hopefully you’ve had some time to think about that, and maybe even come up with your own way of doing it. But how would I do it? And how exactly does this puzzle apply to the larger world of calculating probabilities? Stay tuned because those are precisely the things we’ll be talking about today.

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Should You Go to the Beach?

There are lots of factors that influence your decision about how to spend a day. For example, if you’re contemplating a trip to the beach, you might think about how likely it is to be sunny or foggy, breezy or calm, and if you’ll be able to find a place to park once you get there. Whether or not you realize it, when making decisions like this it’s natural to think about such factors probabilistically. After all, weather reports say things like “there’s a 70% chance of sun after 11am,” or “there’s a 40% chance of rain after 2pm,” and so on. In other words, weather is reported using the language of probability.

And something like how likely it is you’ll be able to find a place to park is most easily thought about in terms of probability, too. For example: Do you have a 50/50 shot of finding parking? Or perhaps you think it’s pretty much a sure thing—something like a 90% chance? Or is it a busy holiday weekend where you’re guessing you have only a 10% probability of being able to park?

The point is that many factors that influence your decisions can be translated into probabilities. Which means that we can use these factors to draw a probability tree to help us make decisions. As I said earlier, this is actually something many of us do naturally without really realizing it…which is great when making simple decisions like going to the beach. But for more complicated decisions, going the extra mile and drawing a probability tree to help you visualize and understand the factors is extremely helpful.

A Probability Tree for the Beach

So, what does the probability tree look like for our hypothetical beach trip? In truth, it could look a lot like the probability trees for coin tosses we’ve been drawing the last few weeks. But instead of starting a new branch at each toss, we instead start a new branch for each of our deciding factors. For example, if we’re interested in whether or not it’s going to be sunny and if we’ll be able to find a parking spot, our probability tree will begin by branching into the possibilities that it will be sunny or cloudy, and then it will branch out from each of these options into the possibilities that we will or will not be able to park.

The top branch represents the possibility that it will be sunny and that we’ll be able to park. If we estimate that the probability of sun is 70% and the probability of being able to park when it’s sunny is 60%, then (as shown in the top-right of the drawing) the total probability for these two things is 0.7 x 0.6 = 0.42 or 42%. The next branch down shows that there is a 0.7 x 0.4 = 0.28 or 28% chance of sunshine but, sadly, no parking spaces for us to be able to get out of the car and enjoy it.

Not All Events Are Independent

While the probability of being able to park when the sun is shining is 60%, our probability tree shows that our estimate of the probability of being able to park when it’s cloudy goes up to 90%. Which means that the next branch of our tree shows us that there’s a 0.3 x 0.9 = 0.27 or 27% chance of encountering lots of clouds but ample parking, and the final branch shows us that there’s a 0.3 x 0.1 = 0.03 or 3% chance of seeing cloudy skies and being shut out on parking.

If you think about this, you’ll notice something interesting about the probabilities of finding a parking spot we’ve used. If it’s sunny, the probability of finding parking is 60%, but if it’s cloudy that probability goes up to 90%. In other words, the probability of one thing in our tree depends upon another. If you think back to tossing coins where each toss is completely independent of another, you’ll realize that this is big news. It’s important to realize that in the real world not all events are independent, and that probability trees are extremely useful for understanding both dependent and independent events.

Using Probability Trees to Make Decisions

Given what we’ve calculated using our probability tree, would you be inclined to take your chances and go to the beach? Looking back on our tree and the fact that we’ve estimated only a 42% chance of sunshine and a parking space, I’m not so sure I would. After all, there’s a 58% chance that we won’t be able to park, that it’ll be cloudy, or—even worse—both! So perhaps we’d be better off making some other plans. But instead of giving up on the beach altogether, why don’t we sketch-up a few probability trees for different days of the week:


One tree (the tree on the left) is the same as the one we looked at earlier, but the second tree (the one on the right) is for the following day. On that day, the chances of sunshine have gone up 10%, and there’s a 30% better chance of finding a parking spot on a sunny day, too. When we put these things together, we find that there’s now a 72% chance of both sunshine and a parking spot…which are definitely odds that I’m willing to take my chances on. So, as we’ve seen, not only can probability trees help you understand the interrelationships between factors in real world problems, they can even help you make decisions!

Tossing Multiple Coins

Before we finish up, I want to leave you with a little assignment that will both help you practice your new probability tree making skills and prepare us to dive into next week’s topic: Imagine flipping 1, then 2, then 3, and finally 4 coins at once. Can you draw probability trees to help you figure out how many possible outcomes there are in each of those cases? And, for each case, can you figure how many of those outcomes have 0, 1, 2, 3, or 4 heads? Once you’ve figured that all out, try to see if you can find a pattern in the numbers. And then be sure to check back next week to learn all about it.

Wrap Up

In the meantime, 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. Finally, please send your math questions my way via Facebook, Twitter, or email at mathdude@quickanddirtytips.com.

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!


About the Author

Jason Marshall, PhD

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.

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