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Below, you'll find an overview of the different probability posts, listed by suggested order of completion:
- An Introduction to Probability
- Five Basic Facts About Probability
- Probability for a Single Event - P(A)
- Probability of Both Events Occurring - P(A and B)
- Optional: Why the P(A and B) Rule Works
- Probability of One or Another Event Occurring - P(A or B)
- Optional: Why the P(A or B) Rule Works
- Probability for Outcomes Not Occurring - P(not A)
- Probability for Outcomes That Are Not Equally Likely
- An Introduction to Combinatorics
- Optional: Why Does the Permutation Rule Work
- Permutations with Repeated Objects
As an LSAT tutor, one of the question types my students most struggle with is “resolve the paradox.” As a law school applicant, the paradox that most nettled me was a paradox centered on law school itself: given that law school is so arduous, why are application figures so robust? Does law school simply attract masochists (or whiners)? Is the well-trod pathway to wealth and political power what tempts people to accept such a hideous fate?
But these questions assume a basic premise, which is that law school is actually hard. Is this true?
Is Law School Hard?
Sorry to disappoint, but the answer to this question is an emphatic “yes”! Of course law school is hard! Have you never talked with a law student? Observe any 1L during finals, or any bedraggled OCI participant, and you will witness the rigors of law school wrought on the human body. Before law school I was youthful and energetic; by November of my 1L year I looked like the “Before” picture in an eye cream ad, and that was on my good days.
Making the decision to attend law school requires accepting that it will likely be difficult.
Why Is Law School (Usually) Hard?
Some people—veterans, parents, interns for Meryl Streep’s character in “The Devil Wears Prada”—don’t find law school particularly onerous because they have survived tougher gauntlets. But most law students are mostly untested in the ways that law schools challenge their students. Looking at the profile of the typical 1L, this is hardly mystifying: a fairly young, relatively recent humanities grad possessed of an abbreviated work history and a roster of academic successes in a context where they are somewhat easy to come by. Despite the maxim that past outcomes do not guarantee future results, most people enter law school either overconfident, underprepared, or both.
A Tale of Two Law School Experiences
One characterization of law school has it like this: you’ll be thrown into a group of dozens of strangers in a pseudo-professional, contentious setting. You will leave behind a lifetime of continuous and easily-won academic validation for a system in which you are evaluated anonymously, for the first and only time, by a professor whose primary interactions with you will be to point out the weaknesses in your reasoning and comprehension in front of the 50+ snickering strangers whose respect you most covet. Unlike the forgiving grading curves of undergrad, which stretched from lowly B+ to unremarkable A+, desirable grades in law school are in limited supply, and you are competing for them against the people with whom you spend most of your waking hours. Remember also that, like you, these people have probably selected into this profession because of a yen for confrontation and an ability to work hard.
But another characterization has it like this: you have to go to class for between 12 and 17 hours per week, with no other responsibilities. To prepare, you will have to read approximately 25-50 pages per class. In the one-in-twenty chance that you get cold-called and can’t remember every single detail of the case, you can just access one of the many overzealous outlines floating around your school and grow comfortable with CONTROL+F and bullshitting—skills that, incidentally, are useful for any attorney.
So Which Experience Will I Have?
The answer to this question is that it’s largely up to you. If you are a welter of insecurities who regards law school success as the paramount test of intelligence and worth, then you might have the former experience. To ensure that the former perspective doesn’t overtake the latter reality, try to keep the following in mind:
- Take law school seriously: it is a professional school, and how well you do will likely shape your career prospects. But that doesn’t mean that you can’t screw up...
- ... And don’t be afraid to screw up: the first time a student in your section gets cold-called and doesn’t know the answer, visceral group mortification sets in. Every time after that point is pretty uneventful, because half the class is on Facebook or shopping online.
- Treat yourself kindly: go to the gym, eat well, avoid forming bad habits, get lots of sleep, have fun, and relax. I learned this lesson the hard way: as a 1L, I took to pouring myself a fluorescent yellow glass of Mountain Dew each night and setting it beside my bed. Why? Because I knew I would be too exhausted to go make coffee in the morning, and I needed a kick to get myself out of bed. I thought, “I can rest when I am a 2L!” until a mentor encouraged me to sleep more regularly and more often, eat better, and exercise. I was doubtful at first, but another paradox that I was happy to resolve was that my comprehension and attitude improved with more leisure and relaxation time and better self-care.
- Maintain your values: decide why you are in law school and stick to it. This doesn’t mean you can’t grow or change your mind, but resist the urge to jump through hoops just because your peers are doing so. Law students slobber over honors like clerkships and law review, but these might not be right for you, might not matter for your career, and might just make you miserable. There are things in life worth suffering and striving for, but make sure they’re important to you before you commit to them.
- Your classmates are going to be your future colleagues, so get to know them through study groups, clubs, and other extracurricular activities.
If you’ve been through law school, what other tips or suggestions do you have? Let us know in the comment section. If you’ve got questions, let us know too.
Conor works as a civil rights attorney for the City of New York, and has been moonlighting as an LSAT tutor for two years. Immediately following law school, he worked as a Ford Fellow at the ACLU’s Women’s Rights Project. He enjoys reading fiction and making bad puns. He is a graduate of the University of Virginia and of Harvard Law School.
As part of a new series here in the 7Sage blog, we've asked our community leaders (Mentors, who were selected from among their peers for their outstanding contributions and character, and Sages, who are community leaders who scored above 170 on the LSAT) to answer a series of questions and provide us with their LSAT wisdom.
This series is just a sampling of the kind of wisdom ready at hand to anyone in our Discussion Forums.
What's the biggest myth about the LSAT?
Mentor Sam “That 2-3 months is plenty of time for a 170+ score. It's not, unless you're a genius or have somehow mastered logic prior to ever being exposed to the LSAT.”
Sage Alex “In my opinion, the biggest myth about studying for the LSAT is that you should only expect to improve your score by 10 points. This claim is not only false, but very detrimental to future law students. Relaying to people that 10 points is all you should expect to gain from studying deters test-takers from devoting enough time to study for the LSAT until they reach their goal score.”
Mentor Brett “In my mind the 2 biggest myths about studying for the LSAT are that you only have to study for 3 months and that you won’t see improvement in any one section. This is a long test, and the skills that it tests aren’t things that you can truly learn and master in 3 months for most people. It’s one where it may take you 3-5 months just to get through the curriculum and then from there you take another 3-6 in taking tests. But the process is worth it in the end. “Also, the assumption that ‘You can’t improve on RC, LG, or LR’ is completely false. Everyone taking this test is different but all of the sections are exactly the same at their base; the questions all test a set of skills and are created by professionals who can exploit the psychology of test takers. All you have to do is learn the skills, avoid the pitfalls, and be able to be able to do this efficiently and confidently.”
Sage Allison “One myth about this test is that it's just an input, output equation with how many hours you put in leading to a particular score. That is part of the equation, but cranking through PTs without good reflection is bad practice. You absolutely must scrutinize your comprehension, question your thought processes, and accept that you have erroneous ways of reasoning that need to be corrected in order to excel on this test. Do your homework, but reflect on your methods and spend time to diagnose your weaknesses.”
Mentor Daniel “That high scores are rare and achievable only among those who are great at tests, geniuses, or savants destined for Harvard. I was once told by an instructor of a test-prep company that I could reasonably expect a score five points higher than my diagnostic. I'm now sitting at an average score increase more than four times that, and I firmly believe at least most should be aiming for a score fifteen-to-twenty points higher than their diagnostic. It takes time, but a score that much higher is doable.”
Ok. LSAT Logical Reasoning—you got this! You're logical. You're reasonable. You destroy (or repair) arguments all the time on Twitter or Tumblr. You've even done some debate in high school or college. How hard can it be?
And then you take your first PT after completing the Core Curriculum.
First few questions are a little wordier than you'd like, but you feel like you got this. You get to question 4 and ...
... Pff ... You know you got this! Just had to get up to speed, that's all. Things are going fine until ...
... You come to a Necessary Assumption question with a really unattractive answer choice that just nags at you. Why did they even bother putting that one in there? And then there's this other answer choice that sounds like EXACTLY what the argument needs ... But is it the right answer choice? And then ...
... A Most Strongly Supported question with an answer choice that seems to be just soft enough, just specific enough, just irrefutable enough to fit the bill for the right answer choice. It's got all the hallmarks of a right answer choice for MSS. So ... You ... Slowly ... Circle ... the AC ...
... And run smack dab into a Parallel Flaw question that takes up the entire left hand side of the page. So you find the flaw in the stimulus ... And then you try to remember if you're supposed to map out the logic in the Answer Choices ... Or is that for the other Parallel question type? You thought you HAD this ...
... And even though you're on FIRE with the next 3 questions, finding those main points, honing in on those flaws in the support structure, naming those assumptions, you're still thinking about that question 2 pages back.
You finish the PT and you question your whole existence for a good 10 minutes.
Then you remember ...
... You've got dreams to grab ahold of.
So you pick yourself up and get ready for some Blind Review.
And you think to yourself ...
And maybe it would.
You worked hard in undergrad and now you're getting ready to start applying. But do you know how hard is it to get into law school? Or more importantly, how do you get into a good law school?
The one thing you need to know about how to get into law school
The answer can be summed up in four letters. LSAT. You need to demolish the LSAT. That's the one thing you need to know.
In the topsy-turvy world of law school applications, LSAT is king.
Isn’t GPA / Personal Statement / Recommendations / Whatever More Important for Getting into Law School?
What about GPA? First, your GPA is pretty much set. Even if you still have another year of grades before you send in your applications, the A in GPA will ensure that the impact of your best efforts won't have much of an impact. Secondly, even though most people agree that GPA is the second most important admissions criteria, it is not nearly as important as the LSAT. A rule of thumb many students use is +1 LSAT point = +0.1 GPA. It's reasonably common for students to improve 10 points on the LSAT with 4 months of studying. Good luck bringing your GPA from 3.3 to 4.3 with 4 months of studying :D.
What about Personal Statements, Recommendations, Extracurriculars, Job Experience and Interviews? They make a difference, but not that much. If you have a lame-duck recommendation or a douchey personal statement, it can tank you. If you were the President of your home country it'll really help.
Most of the time these aren't going to make a big difference. At least not compared to the LSAT. Most of the time, you should put effort into making these shine only after you've taken the LSAT.
WTF? Why do law schools care so much about the LSAT?
There are some obvious reasons, and at least one non-obvious one.
Among the obvious reasons is that the LSAT isn't subject to grade inflation/deflation and competitiveness of different colleges. For example, are a 4.33 GPA from Greendale Community College and a 4.33 GPA from MIT equally impressive? Probably not, at least not academically :) The LSAT acts as an equalizer.
Another reason is that the LSAT tests abstract logic and reasoning, as well as time pressured reading comprehension skills. Both of these are extremely important in law school when you grind through endless readings and try to pull out the arguments and implications.
Here's one non-obvious reason: US News & World Report Rankings.
They rank 200 or so US law schools using a bunch of metrics. One of these metrics is the LSAT. The better the median LSAT score of a school's students, the better the school's rankings. The better the school's rankings, the higher the prestige. More prestige lets the school attract better law students (prestige is like crack for law students), and get a higher median LSAT. The circle of LSAT continues.
If you want to go to a good law school, now you know how to get in. You study your ass off for the LSAT. Sign up for a free trial to get started on LSAT prep. Or jump right in and prep for LSAT with a full 7Sage Course (there is a 14 day money-back guarantee in case you change your mind).
If you're taking practice LSAT PrepTests, then you need to simulate the test environment. It is crucial so that you are ready for the real thing.
It’s really important to experience testing with a simulated proctor so you’re not thrown off on the test day by a person announcing a five minute warning or by the lack of time between the first three sections. In a test that’s as psychological as the LSAT, practicing dealing with those things is critical. - Robyn B.
If you're using our video proctor with real LSAT instructions, that's a great start! But setting up an LSAC approved timer, and finding a place with just the right amount of ambient noise can be a bit of a pain...
So we made an LSAT Proctor App for iPhone/iPad/iPod just for you ;) We designed it to be easy to use, while including all the features we knew were important from teaching thousands of LSAT students. Simulate the LSAT test environment perfectly, anytime, anywhere.
This includes procedurally generated distraction noises, turbo mode, realistic virtual timer, five minute warnings, real instructions, and more. If you like it, give us a great rating! It'll encourage us to keep making great LSAT tools available for free :D
For those of you taking the upcoming June administration of the LSAT or thinking ahead to future administrations, we'd like to share a few best practices/pro-tips to help ensure that you're in top shape heading into the exam. We've included some guidance for the week leading up to the exam as well as for Game Day itself.
Right up front, we'd like to say that you're not going to learn anything new the week before the exam. The hay is in the barn. You've already done the work that will carry you into the exam. Don't cram PT's; at most, do a few sections to keep your mechanics sharp. You need to make sure that you're fresh and in the right mindset for Game Day.
1) Between today and Sunday, go to bed and wake up at the same time every day (and this should be the same time you'll need to wake up for the June exam). Waking up ~3 hours before the earliest time you're likely to start the test (as soon as 30 minutes after the show-up time) will help ensure that your cortisol levels are up and that you're fully awake. Waking up at this time during this week and Monday June 6th helps to ensure that you'll be tired enough to go to bed Sunday night. Also, no screens/blue light after 10pm. This will help ensure that you're not artificially stimulating cortisol (waking yourself up) before bed.
2) Pre-hydrate. Drink a 3-4 liters of water every day of the week before the test. It's really not that big of a deal to drink that much water, and doing so will ensure that you are well hydrated the morning of without having to drink much (if any) liquid.
3) Practice your game day routine at least twice. This means wake up at the time you'll wake up on Monday, eat the exact same breakfast/lunch you plan for game day. Keep track of what you eat and drink and when you do it. Track your hunger, thirst, and bathroom need levels (just like in The Sims). Pro-tip: if you need to go at 2PM, there's a very strong likelihood that if you follow the same plan/timing, you will need to go in the middle of a section. Which is what we want to avoid.
4) Day of, general: Don't do anything differently from your dress rehearsals. No magic pills. No extra coffee. No tricks. No surprises. Perhaps get to the test center early and just go for a walk around the grounds if feasible. You might see some very nervous folks in crisis mode. Disregard. You are not them.
5) Day of, warm up: Whatever you do, don't score anything. And don't do any new material. Maybe take a handful of LR Q's, maybe one easy game, maybe one easy RC. Just chill out about it. You're just warming up your mechanics.
6) Day of, during the break: People will try to talk to you because they are nervous or want reassurance. You are not there to be anyone's friend. You are not there to be anyone's therapist or life coach. However you put up your personal "Do Not Disturb" status—just don't let anyone throw you off your game.
Not sure what all those letters and arrows are that people draw for the LSAT? Watch this video on Sufficient and Necessary Conditions, fundamental to LSAT Logic.
Drawing LSAT Sufficient And Necessary Conditions
You've probably heard a lot about sufficient and necessary conditions on the LSAT. They're tough to get a handle on at first, but not that difficult once you get the hang of it.
A sufficient condition is "enough" to tell you that something else is true. Suppose I tell you that a CEO of a fortune 500 company is powerful.
Then if I tell you that Tim Cook is CEO of Apple, a fortune 500 company, you know something. Tim Cook is powerful.
A necessary condition is something that has to be true, when something else is true. "Powerful" is the necessary condition of the statement I just told you. If you find out that Marissa Mayer is CEO of Yahoo, then you know she is necessarily powerful. It can't be any other way.
A conditional statement has a sufficient and a necessary condition. A conditional statement is true 100% of the time.
So if I tell you that all Fortune 500 CEOs are powerful, don't look for an exception. Just assume that it's true. This isn't a good idea in real life, but it's what you have to do for the LSAT.
Drawing LSAT Conditional Statements With Arrows
It's complicated to try to keep track of several conditional statements. And LSAT logical reasoning questions often give you several conditional statements. So you should use a system of shorthand notation to represent them.
The system everyone has settled on has letters and arrows. Take the statement I gave above. Here's a good way to draw it:
C --> P
I prefer to stick to one letter, or two at most. Some people will try to add more letters, like this:
CEOF500 --> Pow
That quickly gets confusing. The letters should serve as a reminder of the statement, but they don't have to mirror every part of it.
Joining Conditional Statements To Form Deductions
If you have multiple conditional statements, you can often join them. Anytime the necessary condition of one statement matches the sufficient condition of another, you can put them together.
"Every CEO of a fortune 500 company is powerful."
"Everyone powerful is a little arrogant"
C --> P
P --> A
C --> P --> A
"Every CEO of a fortune 500 company is a little arrogant"
You could also draw this as C --> P --> LA, if you prefer to include the "little" in the statement about "arrogant".
Did you like this lesson on LSAT conditional statement diagramming? If so, you'll enjoy our online LSAT course, which has complete lessons on LSAT logic, applied to real LSAT questions.
19 common, repeated argument flaws that students overlook on Logical Reasoning
Note: This list is not meant to be exhaustive. It is simply meant to be a guide to the more common argument flaws the LSAT tends to exploit.
1. Attacking the source of the argument
To attack an argument you may attack:
1) the premises (which never happens on the LSAT); or
2) the support the premises give to the conclusion
What you DO NOT get to do is to attack the author, his past acts or arguments, his motivation, where the argument comes from, or anything other than (1) or (2).
2. Uses terms unclearly/equivocation
The author uses a term (with more than one meaning) inconsistently. For example, “public interest” in one sense means what is in the best interest of the public (e.g., clean air, roads, schools). In another sense, it means what the public is interested in (e.g, celebrity gossip). The shift in word meaning will often be subtle and hard to notice.
3. Analogies that really aren’t analogous enough
All arguments by analogy fall apart at some point. At some point the two things being analogized lose their relevant similarities and the analogy cannot continue. We can say attacking the LSAT questions is like attacking enemy starships. But, in many ways, it’s not.
4. Appealing to authority in an area outside their expertise
Appealing to an authority where the subject matter is outside the expertise of the authority. For example, a dentist’s opinions on automotive maintenance is not authoritative.
5. Causation confusions
Whenever the LSAT concludes or assumes that A causes B, 99.9% of the time it’s wrong. They’ll tell you A is correlated with B or that A coincided with B and therefore A caused B. Maybe. That’s just one possible explanation for the correlation. Here are the other 3 possible explanations:
1) B caused A
2) C caused both A and B
3) A and B are merely coincidentally correlated and really something else, X, caused B. We see this a lot with accident rate and speed sign questions. New speed limit sign was put up! Accident rates drop! Therefore, it must be that the new speed limit sign dropped the accident rate. Maybe. But maybe there was an increase in cop cars patrolling the area and that’s what actually caused the drop in accident rates. The speed limit sign was just coincidentally there.
6. Circular reasoning
Assuming what you’re trying to prove. The premise is a mere restatement of the conclusion.
“Everything I say is true. This is true because I said it, and everything I say is true.”
7. Confusing necessary and sufficient conditions
The oldest trick in the book.
8. False dichotomy
A false dichotomy only pretends to divide the universe into two binary halves. It is not a real contradiction. Consider this real contradiction: cats and non-cats. That’s cleanly cuts the universe into two halves. Garfield? Cat. Einstein, MacBook Pro, Love? Non-cat. Here’s a false dichotomy: Cats and dogs. See how that leaves out Einstein, MacBook Pro, and Love? They are neither cats nor dogs.
9. Confusing probability for certainty
Could be is not must be. Even if something is 99.99% likely to happen, it does not mean that it will happen.
10. Confusing is for ought
Don’t confuse the descriptive for the prescriptive. Descriptive simply describes the state of the world. The tree is small. The lake is murky. Prescriptive reveals values. The tree ought to be big. The lake should be clear. The prescriptive reveals what we care about. You will typically encounter a descriptive premise leading to a prescriptive conclusion. For example, the house is on fire therefore we should put the fire out. That’s not a good argument. There are a number of reasons why we wouldn’t want to put the fire out. We always need a bridge premise to take us from the descriptive world of the premises to the prescriptive world of the conclusion. The bridge in the example argument above would be: Houses that are on fire ought to have their fires put out.
11. Percentages v. quantity
Percentages don’t necessarily reveal quantity and vice versa.
For example, Group A wants a 10% raise and Group B wants a 50% raise. Who will earn more money afterward? Who is asking for more money? We have no way to know based on this information.
12. Surveys and samplings to reach a general conclusion
Remember that surveys and samplings must be random (that is, non-biased). Asking a group of 20 year olds about who they are voting for will only tell you who 20 year olds are voting for (assuming they’re a statistically random set of 20 year olds regarding race, gender, etc.), not who the entire country will vote for.
13. Hasty generalization
Hasty generalization is very similar to sampling error. The difference is that the conclusion is very broad. You cannot make a generalization based on small sample size or based on one or two incidents.
14. Experiments to reach a general conclusion
Experiments to reach a general conclusion must include a control group. It must also establish the baseline of what is measured before the experiment begins.
15. Your argument fails therefore the opposite of your conclusion must be true
Be careful of arguments that try to do this. Just because you’ve wrecked someone’s argument, doesn’t mean that you get to conclude the opposite of his conclusion. If I make a crappy argument for going to the movies tonight as opposed to going to a bar or doing any number of things, you can’t just show me why my argument sucks and conclude: therefore we should go to a bar. First of all, there could be other arguments made to support going to the movies. Additionally, you still have the burden of making an argument that proves that we should go to the bar.
16. Relative v. absolute
A is faster than B, therefore A is fast? Not necessarily so. A is faster than B in relative terms. It doesn’t imply that A is fast in the absolute sense. For example, we know that the conclusion in this statement is not true: “Hippopotamuses are smaller than an elephants. Therefore, hippopotamuses are small.” Or take this statement: “Turtles are faster than ants. Therefore, turtles are fast.”
17. Confusing one possible solution for the only solution
There are many ways to solve a problem. Just because one solution solves a problem doesn’t mean that particular solution is the only solution that can solve the problem. Nor, for that matter, does it mean it is the best solution. If I nuke Cleveland, I’ll probably solve Cleveland’s homeless problem. This does not make it a good solution.
This flaw can also be used in the negative. This happens when one solution to a problem turns out not solve the problem, and then the conclusion might say that the problem cannot be solved or that the problem shouldn’t be solved. The flaw remains: just because one solution to a problem is inadequate doesn’t mean that the problem itself cannot be solved.
18. Red herring
This happens when the argument doesn’t address the relevant issue. Rather, it addresses some other issue that is tangential or has nothing to do with the relevant issue but, for some reason, commands your attention.
19. Tradition fallacy and novelty fallacy
The fact that something is old doesn’t mean that it is right or better. In the same vein, just because things have been done a certain way for a long time doesn’t mean that it is right or better. See slavery.
Likewise, just because something is new doesn’t entail that it is best course of action. Nor does it entail that the old thing or idea is no longer relevant or true. Change for the sake of change is not an argument; there must be something that shows the change is better.
Did you like this list of common LSAT argument flaws? You'll find more information like this in our online LSAT course, along with walkthroughs of hundreds of flawed LSAT arguments.
Sometimes it can feel like we are prisoners to our own habits, and indeed, research suggests more than 40% of our daily actions are automatic processes we no longer screen. As author Charles Duhigg explains in his popular book The Power of Habit, "When a habit emerges, the brain stops fully participating in decision making. [...] So unless you deliberately fight a habit--unless you find new routines--the pattern will unfold automatically." For the LSAT, eliminating bad habits and developing productive new ones can be tremendously helpful in improving your score, especially if you're shooting for the 90th percentile and above.
Three elements make up a habit--the cue, the routine, and the reward. Duhigg suggests that one way to replace a bad habit is to change the routine, as smokers often do by substituting gum or other snacks for cigarettes. Another way to tweak an existing habit is to tack on a new, good habit to the routine, such as doing a few squats while brushing your teeth or more pertinently, adding the blind review process to your normal practice test schedule.
What Duhigg calls "keystone habits" should be another point of focus. These are the habits that when changed, can also impact other habits in positive ways. Making one's bed in the morning is one such habit, producing a small "win" early on in the day and instilling confidence that "bigger achievements are within reach." For the LSAT, you can create a pre-test routine of several "keystone habits" to build up confidence with each step--a process that, as Duhigg writes, will help you feel victorious even before you reach the main event of the day.
Although I wasn't aware of it at the time, I built a pre-test routine similar to what Duhigg recommends, practicing it each time I took a sample exam: eat some oatmeal while reading Wired (stimulating but not rocket science), do a really easy logic game to get my brain going, and "free write" my anxieties (i.e. what could be the worst case scenarios, what would be my plan, and reminders that these things had never happened in the previous practice tests). If nothing else, it cut down on my anxiety by making the exam less of an event and more of a natural next step on my checklist.
You can read more about changing habits in the Power of Habit: Why We Do What We Do in Life and Business.