Probability Theory, the LSAT, and You (Part 2)
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Welcome back to Nerdville, folks! Today, we continue our discussion on the relationship between probability theory and the LSAT. If you haven’t read Part 1 of this series, go check it out.
All refreshed? Have you solved the cookie jar word problem yet? Are you wondering how in the world Bayesian probability theory could link up to LSAT test-taking strategies?
Wonder no further! Let’s begin.
The Traditional Way
In Part 1, I mentioned “falsificationism,” a central tenet of the scientific method for many. It’s the idea, developed by philosopher Karl Popper, that scientific theories have to be “prove-wrong-able” in order to actually describe the way the world works. No scientist can make up a theory that has no potential for being disproven—because otherwise, you can’t really test to see if it’s true or not.
Take Isaac Newton. You may have heard of him. The guy more or less united the movements of apples on Earth with the movements of planets in space: his scientific discoveries in physics (Newtonian mechanics, y’all) were a groundbreaking discovery that changed the course of human history…
Until Einstein came along (you may have heard of him) and proved that Newton’s equations were just a specific case in general and special relativity. In other words, Newton got it right—up until a point, when his theory stopped working. Then Einstein rolled in, figured out what Newton did wrong, and made an even better theory.
The Traditional Way… LSAT Style
Does falsificationism seem familiar? If we step back a bit, we can see that it’s probably how you look for the right answer on an LSAT question—and that’s perfectly normal.
Example: “Okay, answer choice A. Here we go. This first sentence looks good… Oops! It uses the wrong word here! Definitely not A. What about B? Hm. Tough, but I think it’s right. Let’s see if C can beat it. Nah, B is better for sure…”
And so on. Essentially, you’re picking one right answer apart from four wrong answers.
Is there anything wrong with this?
Not at all—at least, not if you’re an LSAT pro. If you can read each incorrect answer choice and correctly identify everything that disqualifies that answer choice, quickly and accurately, then you’re good to go.
But in my experience, plenty of us will read an answer choice and say to ourselves, “Uh, I’m not sure if this word choice means this answer is wrong or not…” It’s not always clear what disqualifies an answer choice and what is merely re-wording! When looking for that binary “right or wrong” understanding, we can slow down, get confused, lose time, lose points, and stress out.
Right & Wrong vs. Likely & Unlikely
One solution is to make use of that Bayesian probability theory we learned about.
Instead of trying to disprove answer choices, like Karl Popper, let’s try to determine the probability for our answer choices, like Thomas Bayes.
“Wait, Ben. There’s one correct answer choice and four incorrect answer choices—that’s how the test works. What do you mean about probability? You’re crazy! It’ll never work!”
First, please calm down.
Second, you’re right—there is one correct answer choice. But the LSAT is a tough test: so tough, in fact, that looking for “the right answer” with 100% certainty can actually distract us, both from picking the best answers and from completing the entire section of the test!
Think about it this way. Just because one answer is 100% correct doesn’t mean that we need to be 100% confident that it’s 100% correct. We just need to be confident enough. If we waited around for total assurance, then we’d never finish every question in the section—we’d spend too long rereading the toughest questions and picking apart the trickiest answers.
So instead of going into the answer choices for each question by asking yourself, “Is this right or wrong?”, trying asking yourself, “Is this a likely or unlikely answer?”
Bayesian Inference, Round 2
Alright! We’re going to think about answer choices in terms of probability, rather than certainty.
Next step: How?
By now I know you’ve read the first post in this series, but I’m going to copy and paste a short section from it anyway:
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 […] with each new piece of evidence.
That’s the ethos of Bayesian inference—using Bayes’ Theorem to think about your beliefs as probabilities, and to alter those probabilities based on evidence. Take a second and consider why this relates to our discussion thus far.
Here’s what I think:
If we look for evidence that increases or decreases each answer choice’s probability of being correct, then we gain the advantages of:
- Evaluating each answer choice independently of the others
It’s natural to ask whether A is better or worse than B in the moment, but we want to evaluate our options for their own merit first!
- Getting granular with what makes each answer choice more right or more wrong—thus avoiding the “It just seems right!” explanation that leads to so many lost points
Looking for specific evidence forces us to dig into the words and phrases of each answer choice. You may have a strong intuition for the LSAT’s questions, but if you rely only on that sixth sense, then you’ll find it difficult to improve your understanding of the test.
When you find a piece of evidence for or against an answer choice, you need to figure out how much that evidence weighs and adjust your probability for the answer choice as a whole. Here are some examples:
“Hm, this answer choice uses the word ‘nutrition,’ but the stimulus talked about ‘weight loss.’ I don’t know if that totally disqualifies the answer, but I think it’s less likely to be right.”
“This answer choice says ‘most’ when the stimulus says ‘many.’ It’s a bit stronger than it needs to be… But this is a Sufficient Assumption question, so that may be okay. I’ll leave this part alone for now.”
“This answer choice uses all of the right words… But I don’t think they’re in the right order. Seems tricky, so I’ll mark it as ‘less likely’ and move on.”
“My Conditional Logic prephrase for this question matches this answer choice exactly. I’m pretty certain this is the correct answer.”
Here’s a great idea: annotate your counter-evidence (by underlining it, boxing it, etc.) and then write out a quick and dirty probability next to each answer choice. The more annotating you do on each answer choice, the more skeptical you ought to be.
- Reading suspiciously, like a detective searching for clues, rather than holistically, like me at an art museum trying to appraise a Pollock
Wrong answer choices are usually wrong for multiple reasons. If you’re suspicious of each weird word and unusual phrase, then you’re more likely to sniff out those answers that are just barely incorrect. Thinking about answer choices as combinations of bits and pieces, each worthy of evaluation, can help you peek under the hood of some intimidating problems.
- Comparing the answer choices we haven’t ruled out once we’ve read them all
More on this point in a future blog post, but the gist is that there’s a difference between not being sure which answer is right and not being sure which answer is better, and thinking in terms of probability moves us from the first to the second.
***
Whew—you made it! Yet another long essay on probability theory and standardized tests. But before you say goodbye…
By no means am I saying that thinking in terms of probability is the right way to tackle the LSAT, or even the best way. There are all sorts of approaches you can take to beating this test—and I believe that Bayesian rationality is one way that may be useful to certain people.
At the very least, I hope you’ve found this series interesting and instructive. I’d recommend trying out this perspective and seeing whether it adds anything to your strategies. Either way, I’d love to hear from you in the comments or via email!
The next and final post in this series will take a (briefer, I promise) look at how probabilistic reasoning can help us in a specific use case: skipping questions for high-performers.
See you soon! (Probably!) ?
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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!