How to Calculate Probabilities, Part 1
Learn how to use probability tree diagrams to calculate probabilities. Then tackle a brain teaser using probability trees to decide if you should go to the beach.
In the last episode, we learned how to draw probability trees—which means we’re now ready to learn how to use these probability trees to calculate probabilities. Why would you want to do that? Because, as we’ll see over the next few weeks, you can use these calculations to make more informed decisions—at work, at home, and everywhere in between. You can even use them to help you decide if you should go to the beach! Want to know how? Stay tuned because that’s exactly what we’re talking about today.
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Recap: How to Draw a Probability Tree
Let’s start by recapping where we left things last time. In particular, we learned that probability trees are a tool to help you visualize the possible outcomes of a series of events. For example, imagine we’re going to toss a loaded coin 2 times. As opposed to a normal fair coin that has a 50% chance of coming up heads, let’s say that our loaded coin has a 60% chance of landing heads and thus a 40% chance of landing tails. As we saw last time, we can chart out all the possible outcomes using a probability tree.
To do that, draw a single dot on a sheet of paper to represent the first coin toss. Next draw 2 arrows heading away from this dot representing the 2 possible outcomes of the toss—heads or tails—and label each branch with the
probability of that outcome (0.6 for heads and 0.4 for tails). Now repeat this for the possible outcomes of the second coin toss. In other words, draw a dot at the tip of each of the 2 arrows to represent the next toss, and then draw a pair of arrows coming out of each of these dots to represent the possible outcomes of the second coin toss. Once you’ve labeled each of these new arrows with their probabilities (again 0.6 and 0.4 for heads and tails), you’re tree is finished.
When to Multiply Probabilities
While being able to draw probability trees and visualize all the possible outcomes of a sequence of events is great, it’s what we can do with probability trees that’s really important. One thing we can do with a probability tree is figure out the probability of two particular successive outcomes. In other words, in our loaded coin example, we know that the probability of tossing heads on any given toss is 0.6. But what’s the probability of tossing 2 heads in a row? Or 2 tails in a row? Or heads on the first toss and tails on the second?
To find out, start by taking an imaginary walk along the path in the probability tree that leads from the starting dot to each of your events. The overall probability of this sequence is found by multiplying together the probabilities you encounter along the path. For example, to find the probability of 2 heads, walk along the upper branches multiplying 0.6 x 0.6 = 0.36 to find that there’s a 36% chance of flipping two heads in a row. To find the probability of heads and then tails, walk along the appropriate branches multiplying 0.6 x 0.4 = 0.24 to find that there’s a 24% chance of this combination. How about tails and then heads? Or 2 tails? If you understand how to find the last two probabilities, you should be able to work these out. Once you get them, take a look at my probability tree drawing to check your answers.
Remember, this method for calculating the overall probability of a particular sequence of events works for any probability tree…not just those about coins! And it works for any length sequence of events, too. In other words, you could draw up a giant probability tree and use the exact same technique to figure out the probability of tossing something like 10 heads in a row…or anything else!
When to Add Probabilities
But that’s not all. For example, what if we aren’t interested in finding the overall probability of a single sequence of events (something like “What’s the probability of heads then tails?”), but we’re instead interested in finding the combined probability of two or more sequences (something like “What’s the probability of heads then tails or tails then heads?”). How does that work?
To find the combined probability of two or more different sequences of events that lead to different outcomes in the same column of a probability table, just add the probabilities. For example, we can find the combined probability of “heads then tails” and “tails then heads” by adding their 0.24 probabilities together to get a total of 0.24 + 0.24 = 0.48. In other words, there’s a 48% chance of getting 1 head and 1 tail—in either order—when you flip our loaded coin twice.
The sum of the probabilities in any column of a probability tree is always equal to 1. Why? Because one of the things listed in the tree absolutely must happen—after all, we listed all the possibilities!—and that means that the combined probability of all the possible events in a given column must equal 1.
How to Use Probability Trees to Make Decisions
A good question to ask yourself now is: What does any of this have to do with real life? Of course, coins are real objects so all of this has had to do with real life. But how about something more tangible…like the decision to go to the beach. Can probability and probability trees help you make that decision? That question leads us perfectly to the brain teaser puzzle I’d like to leave you with this week:
Clearly there are a number of factors that might influence whether or not you decide to spend the day at the beach—things like if it’s sunny or foggy, breezy or calm, and if you’ll be able to find parking once you get there. So here’s the question: Can you figure out a way to think about these factors probabilistically? And then can you draw a probability tree that will help you figure out which day of the week is best for your beach outing? Finally, do you foresee any problems trying to do this?
I should say that this is definitely not an easy puzzle to solve—heck, there may not even really be a good solution! I mostly just want you to put a bit of time into thinking about how it might work so that you start to get a feel for the role probability plays in your life. And certainly don’t feel bad if you get stuck…it is a brain teaser, after all. Once you’ve given it a go, be sure to tune in next week to hear what I have to say.
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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!