Probability Theory, the LSAT, and You (Part 1)
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For reasons that are basically too nerdy to explain, I’ve recently gotten interested in probability theory. Specifically, I’ve been looking into something called Bayes’ Theorem (pronounced “bay-zz”), which underpins one way to think about what probability “means.”
Oof.
What does this have to do with the LSAT? Fair question—but I actually believe that Bayesian probability theory can help teach us how to tackle the LSAT in a productive way.
Let’s take a look at what Bayesian probability theory is, how it’s used, and what it means for our general reasoning processes. In Part 2, we’ll look at how Bayesian probability theory connects to the LSAT specifically, and what that might mean for how we take the test.
Bayes’ What?
Reverend Thomas Bayes was a minister, logician, and theologian in England in the early 18th century. His paper that introduced the now-famous Bayes’ Theorem was published posthumously by his friend Richard Price; a few other mathematicians lay claim to the discovery as well, but let’s be honest—we really only care about the name on the box.
So what is this all-important theorem?
First, let’s look at the mathematical equation:
You might remember from school that P(A) and P(B) are simply statements for the probability of A and B. For example, if A = “aliens exist among us,” then P(A) is the probability of that claim. (The truth is out there!) It’ll range between 0, which means there’s no possibility, and 1, which means it’s absolutely certain.
P(A|B) is called a conditional probability, and it’s a bit wonkier. It reads as follows: “The probability of A, given B.” In other words, if I know B is true, then what’s the probability of A? In this way, Bayes’ Theorem calculates the probabilities of things, using evidence for or against those things.
So if A is “aliens exist” and B is “I saw a spaceship last night,” then we’d use Bayes’ Theorem to figure out the probability that, given the probability that I saw a spaceship last night, aliens do indeed exist.
Side note we’ll revisit: We’re dealing entirely with probabilities because, even if I seriously believe that I saw a spaceship, I still should say to myself that I could be mistaken. According to Bayes, beliefs about the world are probabilities, not propositions (either completely true or false).
Does Anyone Actually Use Probability Theory?
Now, this all sounds like overly complicated gobbledygook. What kinds of questions can Bayes’ Theorem actually help us answer?
Here’s a simple one:
There are two jars full of cookies. Jar #1 has 10 chocolate chip cookies and 30 sugar cookies, while Jar #2 has 20 of each. Let’s say Bob picks a bowl and then a cookie at random, and winds up with a plain cookie. How probable is it that Bob picked a cookie out of Jar #1?
That is to say, what is the probability of “Jar #1,” given “plain cookie”? (Email me if you want the answer!)
That’s cool beans—but we need Bayes’ Theorem for important things, too. Statisticians rely on this equation to determine the likelihood of false positives for medical screenings and drug tests, for Coast Guard search-and-rescue missions, and for Netflix’s “Movies You’d Like” prediction system, among many other uses.
So: we’ve got an important equation, probability theory, and cookie jars. If you’re not sure how this could possibly connect to the LSAT, don’t worry. We haven’t talked about Bayesian inferences yet.
Bayesian What?!
I know, I know. You’re taking the LSAT, not the GMAT: we’re not here for math. Thankfully, the equation itself doesn’t matter to us nearly as much as what it implies about how we can think about the world.
Let’s say I have a particularly odd belief about the world: “My brother is a human, not a secret lizard-person.”
Before learning about Bayes’ Theorem, I’d hold onto this belief until I gathered enough evidence to disprove it. That’s how “traditional rationality” works—for those of you into the sciences, you’ll recognize this as “falsification.”
For example, maybe one day I see my brother eat a fly. Hm… Unusual. And the next day, I noticed that my brother had scales and fangs. Bizarre. And finally, I noticed that he had a tail, yellow eyes, and a forked tongue.
At that point, in an ideal world where people change their minds, I’d throw up my hands and say “Alright! I was wrong! My belief has been disproven—turns out my brother is a secret lizard-person.”
That’s pre-Bayes. What about post-Bayes?
When we think about the world in terms of probabilities that go up or down based on evidence for or against our beliefs, I would “update” my initial belief about my brother with each new piece of evidence.
Catch him eating a fly? Weird—but not out of the question. My belief that my brother is not a lizard-person would go from 0.9 to 0.88.
Noticing scales and fangs? Definitely evidence against my belief—suddenly I’d be looking at a belief probability of 0.5.
By the time I see my brother’s prehensile tail, my belief that he’s not a lizard-person will be approaching zero… And the competing belief that he is a lizard-person would be on the up-and-up.
Note that the numbers aren’t too important; they’re basically made-up. What’s important is that we’re thinking in terms of reinterpreting our belief as more or less likely, depending on the strength of our evidence.
You see, instead of thinking about beliefs in terms of “true or false,” you can think about them in terms of “very likely,” “sort of likely,” “unlikely,” and so on. Learning new things will affect your beliefs without forcing you to flip that big switch of “true” to “false.” You’ll also think about competing beliefs, rather than your belief. Making it plural and dropping the possessive will help you change your mind when you need to—you become more concerned with accurate beliefs and less worried about defending “your” position.
This isn’t just a theoretical exercise, either: studies show that quick-learning infants actually use Bayes’ Theorem intuitively, and Bayesian reasoning is helping many fields of science back up or refute the results of questionable studies.
So, About the LSAT…
One long blog post later and nary a word of advice about our favorite legal exam! But never fear: Part 2 will cover exactly how to take what we all now know about Bayesian reasoning and apply this probability theory to our LSAT test-taking strategies.
If you’ve got any ideas about how these topics connect (in my mind, at least), send me an email with your predictions! ?
Don’t forget that you can attend the first session of any of our online or in-person LSAT courses absolutely free. We’re not kidding! Check out our upcoming courses here.
Ben Rashkovich
is a Manhattan Prep LSAT instructor based in New York, NY. He’s a graduate of Columbia University, and he scored a 172 on the LSAT. He enjoys the mental challenge and logical acrobatics of the LSAT—and he feels that studying for the test can teach everyone to approach problems more rationally. You can check out Ben’s upcoming LSAT courses here!