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In our last post, we talked about the idea of an experiment, outcome, and event. If you're not familiar with those concepts, it may be a good idea to look at that post. Here, we will talk about some of the basic features of probability. First, a definition:
Definition: The probability of an event is a number that measures the likelihood of the event occurring.
And because it is tedious to always write out things like "the probability that a fair coin lands leads is ½", we will adopt an abbreviation. We use letters to represent events:
E = A fair coin lands heads
And then, we just write:
P(E) = ½
which we read as:
The probability that “A fair coin comes up heads” is ½.
And in general, for any event E, we use P(E) to denote the probability that event E occurs. This shorthand will save us much space in the rest of the series.
Now, a probability measures the likelihood of an event. This brings us to:
5 Basic Facts About Probability
1. A probability of 0 means that an event is impossible.
So if you find that P(E) = 0, that means that E will not occur. As an example, when rolling a six-sided die, the event that we roll a 7 is impossible -- it does not occur in any of our outcomes. Thus, P(Roll a 7) = 0.
2. A probability of 1 means that an event is certain.
So, when rolling a six-sided die, the event that we roll some number is a certainty -- it occurs in all of our outcomes. Thus, P(Roll a number) = 1.
3. An event with a higher probability is more likely to occur.
So, if the probability that it snows is 20% while the probability that it rains is 80%, then it is more likely to rain than it is to snow. And, on the flip side, events with a lower probability are less likely to occur.
4. Probabilities are always between 0 and 1.
This makes sense, since if an event had a probability greater than 1, then it would be more likely to occur. But events with a probability of 1 are already certain to occur! How could any event be more likely than a certainty? Similarly, if an event had a probability less than 0, then it would be less likely to occur, but events with a probability of 0 are already impossible! How could an event be less likely than an impossibility?
This also gives us a helpful way to check our answers: if we get a probability greater than 1 or less than 0, we have made a mistake somewhere.
5. The probabilities of our different outcomes must sum to 1.
E.g. if we have 4 different outcomes, then
P(Outcome 1) + P(Outcome 2) + P(Outcome 3) + P(Outcome 4) = 1.
This is because, when we do an experiment, something is bound to happen. So the probabilities of our outcomes must sum to 1.
Now, for the GRE, there are three main types of probability problems:
- The probability of a single event occurring: P(A)
- The probability that two events both occur: P(A and B)
- The probability that one or another event occurs: P(A or B)
You are about to do an experiment with four possible outcomes: A, B, C, and D. The stated probabilities are as follows:
P(A) = .5
P(B) = .3
P(C) = .38
P(D) = .1
Is such an experiment possible? What if P(D) = -.18?
Give an example of an experiment not discussed above, and give an example of an event with a probability of 0 for that experiment, and another event with a probability of 1 for that experiment.
Translate P(A) + P(B) = ½ * P(C) into a natural language (like English, French, Chinese, etc.).
No, such an experiment is not possible since P(A) + P(B) + P(C) + P(D) = 1.28 which is not equal to 1. And changing P(D) would make P(A) + P(B) + P(C) + P(D) = 1, but then we would have P(D) = -.18, which is a negative number. Since probabilities cannot be negative, this experiment is again impossible.
There are many possible examples; here is one: investing in the stock market. It is a certainty that "The value of my portfolio will either increase, decrease, or stay the same." It is an impossibility that "The value of my portfolio will both increase and stay the same." It may do one, or the other, but to do both is impossible.
Again, there are many possible solutions. Not confident in my French, I'll stick with giving the English translation: "The probability of A plus the probability of B is equal to half of the probability of C."
You're wondering whether you should go see a new action movie, Muscle Man: How One Man's Muscles Save the World (again). Now, you're not the biggest fan of action movies, but you do enjoy one from time to time. So you figure it'll be worth it if you can get a good seat or if the action sequences are amazing. If neither of those things happens, then it's not worth going for you. Now, you're wondering, should I go see the movie?
Well, something that matters to you is the likelihood of: (i) I get a good seat, or (ii) the action sequences are amazing. Let A = "I get a good seat" and B = "The action sequences are amazing." You really want to know the value of P(A or B). If it's really high, then the movie is probably worth it. If it's really low, then the movie probably isn't (and buying the ticket and so on just isn't worth it).
In this section, we'll talk about calculating P(A or B), which we read as "The probability that A or B occurs." But before we do so, we need to issue an important clarification: A or B means that A occurs or that B occurs or that both occur. In other words, if A and B both happen, then 'A or B' happens as well. This is somewhat at odds with how we often use the word "or," as in sentences like, "You can study hard or you can fail the test," with the implication being that you cannot do both. Excise that meaning from your mind; in probability, we say that "A or B" occurs if A occurs, or if B occurs, or if A and B occur.
Now, in order to calculate P(A or B), it will help to introduce the idea of mutually exclusive events:
Definition: Two events, A and B, are mutually exclusive if it is impossible for them to both occur.
Another way to put this (symbolically): P(A and B) = 0.
Here are some examples of mutually exclusive events:
- When flipping a coin, getting heads and getting tails are mutually exclusive events. It is impossible to get heads and tails from the same flip of a coin.
- Suppose you and your friend have both entered into a raffle that only picks one winner. Then, the event of you winning is mutually exclusive with your friend winning.
- Suppose you are taking a class in college and you need an A or B to graduate with honors. The event where you get an A is mutually exclusive with the event where you get a B.
The point of talking about mutually exclusive events is to make it easier to calculate probabilities of one event OR another event occurring. We can do such calculations via the following rule:
Mutually Exclusive Rule for P(A or B)
Let A, B be mutually exclusive events. Then, P(A or B) = P(A) + P(B).
In other words, the probability that A or B occurs is equal to the probability that A occurs plus the probability that B occurs.
And if you like to get a sense for why such rules work (rather than simply memorize the formula), see here for an illustration that helps make the rule more intuitive. Now, let's see this rule in action:
In rolling a fair, six-sided die, what is the probability that you will get a 1 or a 4?
We know that rolling a 1 and rolling a 4 are mutually exclusive events, since it is impossible for them both to occur. We know that P(Rolling a 1) = ⅙ and that P(Rolling a 4) = ⅙. Thus, by our above rule, P(Rolling a 1 or Rolling a 4) = P(Rolling a 1) + P(Rolling a 4) = + ⅙ + ⅙ = ⅓.
Now, this rule is only a special case of a more general principle. In general, for all events, and not just mutually exclusive ones, the following is true:
General Rule for P(A or B)
Let A, B be two events. Then, P(A or B) = P(A) + P(B) - P(A and B).
I.e. the probability that A or B occurs is equal to the probability that A occurs plus the probability that B occurs minus the probability that A and B occur.
And again, if you like to see why such rules are true, click here. Here is an example of using this rule:
You are wondering whether to go to the cafe. You would go if you knew that Bertrand or Simone was going. There is a 45% chance that Bertrand will go. There is a 20% chance Simone will go, and there is a 15% chance that both Bertrand and Simone go. What is the likelihood that Bertrand or Simone will go to the cafe?
Let B = "Bertrand goes to the cafe" and let S = "Simone goes to the cafe." Then, the question gives us that P(B) = .45, P(S) = .2, and P(B and S) = .15. Following our rule, P(A or B) = P(A) + P(B) - P(A and B) = .45 + .2 - .15 = .5. Thus, there is a 1/2 chance that Bertrand or Simone will go.
In our next post, we will look at a strategy that can help us solve some tricky questions: instead of finding the probability of some event, try finding the probability that it does not occur.
In a bag, there are 5 red marbles, 2 blue marbles, and 1 pink marble. I will pick one marble from the bag, set it aside, and then pick another marble from the bag. Is the event of my drawing a blue marble on the first draw mutually exclusive with drawing a pink marble second? Is drawing a pink marble first mutually exclusive with drawing a pink marble second?
Drawing a blue marble first and a pink marble second are not mutually exclusive; it is possible to do both. But, drawing a pink marble first is mutually exclusive with drawing a pink marble second. After all, since there is only one pink marble and we do not replace the marbles after we draw them, once you draw the first pink marble, you've drawn the only one there is! You can't draw a second pink marble.
Jane is worried that her new neighbor both (i) likes bad music, and (ii) is willing to blare his preferred kind of music at all hours. She estimates the probability that her neighbor likes bad music at .4, and the probability of his constantly blaring music at .3. And she estimates the probability that (i) or (ii) is true at .6. What probability should she assign to the worst possible outcome: her neighbor both likes bad music and is willing to blare music constantly?
This is a different way in which we can apply the rule we just learned. We have been thinking of our rule as a way to calculate P(A or B). But, if you are given P(A or B), P(A), and P(B), we can also use it as a way of calculating P(A and B). Let's plug in the numbers our question gives: .4 + .3 - P(A and B) = .6. Then, subtracting .6 and multiplying by -1 on both sides, we get: P(A and B) = .1.
There are two events, A and B. The probability of just A occurring is r. The probability of just B occurring is s. The probability of neither A or B occurring is t. What is the probability that both A and B occur?
Now, there are four possibilities in total: (1) A and B, (2) A and not-B, (3) not-A and B, and (4) not-A and not-B. We know that one of these four must happen since they capture all the logical possibilities. Thus P(Possibility 1) + P(Possibility 2) + P(Possibility 3) + P(Possibility 4) = 1. The question gives us the probability of (ii), (iii), and (iv). It asks us about the probability of (i). Plugging in the values we get from the question, we get: P(Possibility 1) + r + s + t = 1, and thus, P(Possibility 1) = 1 - r - s - t.
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