The BriefA Blog about the LSAT, Law School and Beyond
The annual uproar about law school rankings might lead you to believe that the rank of the school you attend is the only factor in determining whether you will become a successful lawyer. As Above The Law points out, the T14 law school rankings, as determined by US News and World Report, rely heavily on inputs – especially peer assessment, grades, and LSAT scores — while ATL’s rankings rely more heavily on outputs like jobs and starting salaries. Given that the two lists overlap quite heavily at the top, I wouldn’t blame you for feeling like you might as well say goodbye to your law career before you’ve even read your first case note if you don’t get into a T14 school. But don’t lose heart! Many, many law school graduates attend non-T14 schools and go on to have successful law careers.
I speak from experience. By way of background, I graduated from Emory Law School squarely in the middle of my class. It was a great place to go to school, with whip-smart professors and clinics, but it was not T14 when I attended and still isn’t (though it’s been solidly T25 for many years). Emory is also located in Atlanta, which, for all of its charms, was not the city where I intended to practice upon graduation. Like so many others, I had my eyes set on New York City. I managed to write myself onto the law review which, given my highly mediocre class ranking, definitely helped boost my resumé. This, combined with my comfort with interviewing, helped me land a job in Big Law in the New York office of a Chicago-based firm, where I specialized in real estate law.
I jumped ship after five years and wound up in Cardozo’s admissions office, where I counseled prospective students about whether they should or shouldn’t go to law school, and why they might be a good fit for Cardozo in particular. I later returned to practicing real estate law with the New York City Economic Development Corporation. As a lawyer, first in private practice and later for the City of New York, I regularly interviewed candidates for summer associate and lateral positions. While I can’t speak for every law firm or government agency, I do think I have some insight about whether attending a T14 law school really matters—so here goes!
When does attending a T14 law school really matter?
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
Sometimes, the order of our objects does not matter. For example, suppose a professor is trying to assign people to different study groups. Then, it does not matter who gets assigned to the study group first; what matters is who is in which study groups, as in the following example:
A professor is trying to split up his class of 9 students into 3 groups of 3. How many ways can he do this?
Or we might have:
An appeals court is convening a 3-judge panel to hear a certain case. There are 26 judges in the courthouse. How many 3-judge panels are possible?
Now suppose that of the 26 judges, 13 are conservative and 13 are liberal. How many possible 3-judge panels will have at least one conservative and one liberal on the panel?
To address these kinds of problems, we will use the following:
Rule for Combinations
If you have n objects and you want to pick k of them to form a group, then there are n!/((n-k)!k!) ways to do so.
Let's try to solve our earlier example with this rule.
By applying our rule, we know that there are ways of choosing 3 judges from the total group of 26 judges in order to form a panel. We can directly calculate this:
, since and .
So there are 2600 ways to pick such a panel.
But why does this rule make sense? Well, let’s start with our rule for permutations: if I want to form a line of three judges from this group of 26, then there are 26*25*24 ways of doing this. This is because we have three spots to fill:
____ ____ ____
And for the first spot, we can pick any of the 26 judges; for the second, we can pick any of the remaining 25 judges, and for the last spot, we can pick any of the remaining 24 judges.
But here is the problem: we don’t actually care about the order in which we pick the judges. Suppose we have Judge Abby, Judge Ben, and Judge Cynthia. Then, we want to treat this possibility:
Abby Ben Cynthia
the same as this possibility:
Ben Abby Cynthia
which is in turn the same as
Cynthia Abby Ben
Cynthia Ben Abby
Abby Cynthia Ben
Ben Cynthia Abby.
We want to count all of these possibilities just once, since for us, a panel with Abby, Ben and Cynthia on it is just the same as a panel with Cynthia, Abby, and Ben on it. They’re not actually different!
Now, whenever we pick three judges from the 26 total judges, there are 3! ways to order those judges. This is because, for the first slot, we can pick any one of the three judges, for the second slot, we can pick any one of the remaining two judges, and for the last slot, we are stuck with the final judge.
So our permutation 26*25*24 actually counts each of these panels 3! times, when we only want to count each panel once. Suppose a panel has judge A, B, and C on it. Then our permutation counts:
So since our permutation is counting these panels 3! times when it should be counting them just once, we divide by 3! to get our final answer.
And, in general, the formula for how many combinations of k items we can pick from n objects is just equal to the number of permutations of k items we can pick from n objects, divided by (k!). Dividing by k! is how we make sure we are not counting all the different orders a particular combination can be put in.
A professor is trying to split up his class of 8 students into two groups of 4 students each. How many ways can he do this?
Notice that if we pick one of the groups of 4, then the other group is already chosen (just as the people who are left out of our initial group). Thus, we just need to determine how many ways the professor can pick a group of 4. Following our rule, there are ways to do so.
Congress has decided to randomly pick a group of 4 congresspeople to lead a congressional committee. There are 535 congresspeople. How many groups are possible? What is the probability that any particular congressperson will be on the committee?
This is fairly straightforward: how many ways to pick 4 people from 535 people are there? That is just ways. Now, to determine the probability that any particular congressperson will be on the committee, suppose that person is on the committee. Then, we must pick 3 other people from 534 remaining people. There are (following our rule) 25236484 ways to do so. Thus, there is a likelihood of that any particular congressperson will be on the committee.
Congress has decided to randomly pick a committee of 4 congresspeople, but they decide to pick as follows: 2 seats are randomly chosen from the Representatives while the remaining 2 are randomly chosen from the Senate. How many committees are possible? What is the probability that any particular Representative will be on the committee? What is the probability that any particular Senator will be on the committee?
Now, we need to find the two parts of the committees separately and then multiply them together. So, for the representatives, there are ways to choose while for the senators, there are ways. Thus, there are total ways to choose the committees. Note how much smaller this amount is than before. And for any particular representative to be on the committee, that person must be one of the two chosen from the representatives. Fixing one of the representatives allows 434 options for the other person. Thus, there is a chance of any particular representative being chosen. And fixing one of the senators allows options for the other person and thus, chance of being chosen.
And finally, we want to address the case when we have permutations with repeated objects. What if we have something like:
The DMV is wondering how many new license plates it can form from just the letters: T, T, and R. How many such license plates are there?
Now, this question is something of a hybrid between our permutations and combinations. It is true that the order of the objects matters: TTR is not the same license plate as RTT. But we can’t just treat it as a standard permutation where we calculate n!/(n-k)!. If we tried that, we would get:
3!/(3-3)! = 3! = 6
But, if we list out all the possibilities, we find that we cannot form 6 different possibilities. The only possible license plates are:
You can try to think of another ordering, but there isn’t one. So treating it as a normal permutation leads us to the wrong answer. What is going on here?
To see what’s going wrong, let’s label our two T’s as T1 and T2. Now, let’s see how many ways we can order them:
T1 T2 R
T1 R T2
T2 T1 R
T2 R T1
R T1 T2
R T2 T1
Now we get the six possibilities we were expecting. But, remember, as a license plate, it doesn’t matter whether one uses T1 or T2. At the end of the day, the letter that shows up on the plate is just T. So, for example, combinations like R T1 T2 are just the same as R T2 T1. They both result in license plates that look like:
R T T
So, like in our combination problem, our permutation is over-counting things. It is treating as distinct some possibilities that are, in fact, the same. Here, our permutation gives us 6 possible outcomes. But, in fact, the ones with the same color are the same:
T1 T2 R
T1 R T2
T2 T1 R
T2 R T1
R T1 T2
R T2 T1
And we see that, in fact, we only have 3 different possibilities, which is what we found in listing out the possible license plates.
So our rule for these cases:
Permutations with Repeated Objects
Suppose you want to order k objects where one of the objects repeats n times. Then, the number of possible orderings is: .
is again a way of correcting for our over-counting. In our case, we can think of this rule as saying: there are 6 permutations. But within those permutations, the order of the T1 and T2 doesn’t matter. There are 2! ways to order T1 and T2, so the total number of distinct ways to order R, T, and T is just: 6/2! = 3.
Here is an application of this rule that involves larger numbers:
You are playing Scrabble. Your hand has: T, R, S, S, S, E, and F. How many possible ways are there to order all of your tiles?
Following our rule, we want to find a way to order 7 objects (i.e. our 7 letters) where one of the objects is repeated 3 times. Then, there are 7!/3! ways to do this. And again,
7! = 7*6*5*4*3!
7!/3! = 7*6*5*4 = 210
There are 210 possibilities
You are trying to come up with anagrams. You get the letters: A, Z, P, P, E, D. How many 6-letter combinations can be formed?
Following our rule, we get combinations.
You are playing Scrabble. You get the letters: S, S, S, S, T, E, P, X. How many 8-letter combinations can be formed?
Following our rule, we get combinations.
So far we have talked about calculating probabilities of events where we tell you all the possible outcomes. For example, when we flip a coin, the possible outcomes are known: heads or tails. In such examples, you know how many possible outcomes there are. If we roll a six-sided die, there are clearly just six outcomes. But what if we are not given the number of possible outcomes? Consider this problem:
There is a bag of 5 marbles; 3 blue and 2 red. Jane picks one marble out of the bag, looks at its color, sets it to the side (and does not return it to the bag) and picks another marble. What is the probability that she picks two blue marbles?
In this case, it isn’t clear how to apply our earlier rules. Remember our first rule:
Probability of Equally Likely Outcomes:
If you have n possible outcomes, all of which are equally likely, then the probability of any particular outcome occurring is 1/n.
How do we apply this rule to Jane’s situation? How many possible outcomes do we have?
This is where combinatorics comes in: it will allow us to determine how many possible outcomes we have. Now the word “combinatorics” is a little intimidating, but it’s just a fancy word for the mathematics of counting. In this case, we will use it to count the number of possible outcomes for Jane.
Let’s try to list out all the possible pairs of marbles that Jane could pick. To do this, let’s number our blue marbles 1 through 3, and number our red marbles 4 through 5. Now, Jane has to pick two different marbles. So, in other words, Jane ends up with some pair of marbles from the following list:
So these are all the possible outcomes Jane could get. She could pick any of the 5 marbles first, and then she could pick any of the 4 remaining marbles second. Now, the question doesn’t say this explicitly, but we will assume (and it is generally fair game to assume) that Jane is equally likely to pick any of the marbles from the bag. After all, the marbles are presumably the same size and feel the same, so she’s just as likely to pick one as to pick any other.
Now, we have our set of equally likely outcomes. And these are all the outcomes. So, in line with our above rule, to find P(Jane picks 2 blue marbles) now we just need to pick out the outcomes where Jane picks two blue marbles. And that’s easy enough to see on our table, we just need to pick the outcomes where two blue marbles are chosen.
Adding up these outcomes, we get 6/20 = 3/10. So that’s the likelihood of Jane picking two blue marbles.
In general, it will not be practical to make our table of possibilities and add things up. For if Jane had a bag of 10 marbles, we would have had 90 possible outcomes (10 choices for the first marble multiplied by 9 choices for the second). So, we introduce the following rule:
Rule for Permutations
Suppose we have n objects that we want to fill k spots where k ≤ n. Then, the number of possible ways ("permutations") to do this is:
The n! is read as “n factorial” and it means n * (n-1) * (n-2) * … * (1). Basically, we take n and we multiply it by all the positive numbers less than or equal to it. Here are some examples:
And, by a special convention, we say that 0! = 1.
Now, to solve probability questions, we want to find: how many permutations (or possible outcomes) will satisfy our event, and how many permutations in total are possible (i.e. how many possible outcomes there are). So in our above problem, we want to find:
- How many permutations involve two blue marbles being drawn? (How many outcomes involve two blue marbles being drawn?)
We have three blue marbles, and we are drawing two marbles. So we want to use three objects to fill two spaces. Following our rule, we get:
- How many total permutations are there? (How many possible outcomes are there in total?)
We have five total marbles and we are drawing two of them. So we want to use five objects to fill two spaces. Following our rule, we get:
Thus, we get: P(Draw two blue marbles) = , which matches our previous answer.
Suppose you have three friends: Anne, Beth, and Charles. How many ways can they form a three-person line?
Following our rule, we want to fit three friends into three places. Thus, there are ways to do this.
Mark is trying to descramble a series of letters: AQCFXDE. How many possible five-letter combinations can be formed from those letters?
Following our rule, we want to fit seven letters into five places. Thus, there are ways to do this.
Jane is drawing four cards from a standard 52-card deck. What is the probability that she first draws the King of Hearts, and then the Ace of Spaces, and then the Jack of Clubs, and finally the Three of Hearts?
We want to answer: how many ways can she draw those four cards in that order, and how many ways can she draw four cards in general. Clearly there is just one way to draw those four cards in that order. Now, how many ways are there for her to draw four cards in general? We are essentially trying to fit 52 objects into four slots, so our rule tells us that there are ways to do so. Thus, the odds are