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A Blog about the LSAT, Law School and Beyond

Thus far, we have mostly talked about problems where all of the outcomes are equally likely. So, for example, our coins are equally likely to land heads as they are to land tails, and our spinners are equally likely to land on any of the outcomes, and our die are equally likely to land on any of 1 through 6. But not all GRE problems are like this. Consider the following:

Example 1
Suppose you have a weighted coin which is twice as likely to land tails as it is to land heads. You are about to flip this coin. What is the likelihood of it landing heads?

In these problems, we need to do a little bit of algebra to find the answer. Feel free to think about it, and if you want to see a solution, click below:

Answer

So in questions where the outcomes are unequal, we can use variables as placeholders for the true probabilities and then use the probability rules to derive the actual probabilities of the different outcomes. Here are some practice problems that are basically variations on this theme:

Practice Problems

Question 1
Suppose you have a weighted six-sided die that is four times as likely to land on the number 4 as it is to land on any of the other numbers. What is the probability that it lands on 6?

Answer

Question 2
Suppose you have a company raffle. Managers (of which there are 30) get 2 tickets, ordinary employees (of which there are 110) get 1 ticket, and executives (of which there are 5) get 20 tickets. What is the probability that the winner is an executive?

Answer

Question 3
Suppose you have an 8-sided spinner numbered 1 through 8. Now, your spinner is set up so that you are twice as likely to get 2 as you are to get 1, 3 times as likely to get 3 as your are to get 1, and so on for all the other numbers up until 8. What is the probability that you spin a 1?

Answer

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Now, we discuss a trick that helps in some problems. Sometimes, instead of finding the probability of an event, it can be easier to find the probability that the event does not occur. Since we know that either an event will occur or not, we know that the probability of the event occurring plus the probability of the event not occurring must equal to 1. More formally, we write:

Probabilities for Complements
Let A be some event. Then, P(A) + P(A does not occur) = 1.

Here's an example where we apply this rule:

Example 1
You are about to roll a fair six-sided dice. What is the probability that you roll a prime number?

Answer

Why do we bother with this rule? Sometimes it is actually more convenient to calculate things this way. For example,

Example 2
You are going to roll a fair six-sided dice 10 times in a row. What is the probability that you get a six at least once?

Answer

Now, it will often be easier to just directly calculate the probability of a given event. But if that calculation looks absurdly difficult or tedious, take a minute to step back and consider: can I calculate the probability that the event doesn’t occur? Would that be easier? Sometimes that can save you a lot of hassle.

Practice Problems

Question 1
You are rolling a fair spinner with seven sections, numbered 1 through 7. What is the probability that, in 10 spins, you get at least one prime number?

Answer

Question 2
One random employee will be selected to win a company-sponsored vacation. There are 500 employees, 300 of which are female. Also, 50 employees are managers, and half of the managers are female. What is the probability that the winner is not a female manager?

Answer

Question 3
The weather reporters say that there is a 60% chance of rain for each of the next seven days. If that is true, what is the probability that it rains at some point over the next seven days?

Answer

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Welcome to Law School Success Stories, where we discuss 7Sage applicants who made the most of their GPA and LSAT score.

👤 Who: “Sarah,” an applicant who grew up in China and moved to the United States for college.

  • 📈 LSAT: 169
  • 📉 GPA: 3.33

Results:

  • 🏆 Accepted at the University of Michigan Law
  • 💵 $35,000 merit scholarship

🥅 Goals and Strategy

Sarah knew she wanted to take her law degree back to China, and the cachet of a T-14 school was important to her. Her parents, however, had a limited ability to pay for her education, and as a Chinese citizen, she wasn’t eligible for federal loans, so she was also hoping for a merit scholarship. Continue reading

Featured image: Steps by Free-Photos is licensed under Pixabay License.

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On today's episode, J.Y. talks about Main Conclusion (aka Main Point) questions and why they're foundational to logical reasoning.

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Welcome to Law School Success Stories, where we discuss 7Sage applicants who made the most of their GPA and LSAT score. Please note that we changed certain details to protect this applicant’s anonymity, but we did not change his numbers or results.

👤 Who: “Neil,” a recent college grad of Southeast Asian descent

  • 📉 LSAT: Under 149
  • 📈 GPA: Over 3.8
  • 🗞 Two-year résumé gap

Results

  • 🏆 Accepted at a T-14 school
  • ✍️ Handwritten note from the dean: "I loved your essays" (and more)

🌘 The Strategy: A Shot at the Moon

Continue reading

Featured image: Person Standing on Hand Rails With Arms Wide Open Facing the Mountains and Clouds by Nina Uhlíková is licensed under Pexels License.

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On today's episode, J.Y. talks about how to score your preptests to make them better predictors of your actual LSAT score.

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Welcome to Law School Success Stories, where we discuss 7Sage applicants who made the most of their GPA and LSAT score.

👤 Who: “Mark,” a Caucasian male in his mid-forties switching careers from the trucking industry

  • 📉 LSAT: 166
  • 📈 GPA: 3.9

Results:

  • 🏆 Accepted at Northwestern
  • 💵 Significant merit scholarship

🚚 Starting the Journey

Mark worked in the trucking industry for twenty years before he began a new kind of long-haul journey toward his JD. He didn’t know any law school applicants beyond the 7Sage community and had no idea where to begin.

Continue reading

Featured image: White Dump Truck Near Pine Tress during Daytime by 500photos.com is licensed under Pexels License.

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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:

  1. The probability of a single event occurring: P(A)
  2. The probability that two events both occur: P(A and B)
  3. The probability that one or another event occurs: P(A or B)

Practice Questions

Question 1
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?

Answer

Question 2
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.

Answer

Question 3
Translate P(A) + P(B) = ½ * P(C) into a natural language (like English, French, Chinese, etc.).

Answer

Next Article: Probability for a Single Event


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Some GRE questions ask about the likelihoods of different events. For example:

Example 1
You are about to roll two fair six-sided dice. What is the probability that they sum to 7?
A. 1/36
B. 1/6
C. 1/7
D. 1/9
E. None of the above.

In this series, we cover the strategies you need for probability questions on the GRE.

Before continuing, you should know that probability questions make up only about 5% of all math questions on the GRE. That totals to about 2 questions per test. So if you are weak in other areas that appear more frequently (e.g. algebra, fractions/ratios, reading graphs), it might be wiser to look at those topics first and return here later.

What Is Probability?

Sometimes, we are not sure what will happen. It may rain tomorrow or not. I may win the lottery or lose. Federer may win Wimbledon again, or not. Probability is a way to handle this uncertainty. Even if we cannot know exactly what will happen, we can at least determine how likely the different possibilities are. So, if I roll a fair die, I can't know if it will land on 6 or not. But I know that it is more likely to land on an even number than to land on 6 specifically.

Talk about probability is commonplace. We might say it's "pretty likely" to rain later today, or that some team has "no chance" of making it to the playoffs. The mathematical theory of probability gives us a way to make that kind of talk precise, by turning it into formulas and numbers.

Probability begins with the idea of an experiment:

Definition: An experiment is an action that, when performed, leads to exactly one of many possible outcomes.

All sorts of things can be experiments. If I don't study for the final, that is an experiment; it could lead to my acing the class (unlikely), barely passing (more likely), or failing (very likely). In the land of GRE problems, standard experiments include things like flipping a coin (which either leads to the outcome of heads or the outcome of tails), rolling a die (which leads to some number 1 through 6), or picking a winner for a raffle (where the "outcome," or in other words the winner, could be anyone who bought a ticket).

Now, let's consider rolling a fair six-sided die. That experiment has 6 possible outcomes:

File:Dice.png

And there are lots of questions we could ask about rolling a dice. For example: am I more likely to get an even number or an odd number? What are the odds of getting a prime number? To answer these questions, we need the idea of an event:

Definition: An event is a set of some (or none, or all) of our experiment's outcomes.

Here are some examples of events:

  • My die lands on an even number
  • My die lands on a number which, in English, is made of 4 letters
  • My die lands on 6

Note that events do not need to be possible; we can also consider events that simply will not occur:

  • My die lands on 7
  • My die lands on a number that is neither even nor odd
  • My die lands on a number which, in English, is made of 47 letters

Now we are ready to get at the heart of probability: the whole point of probability is to figure out what the likelihoods of different events are. These GRE questions will give you some setup and some event, and ask you to find the probability of that event. And the rest of this series will be devoted to doing exactly that -- figuring out those likelihoods -- across a series of increasingly complicated contexts.

And here are some practice questions to test your understanding of the above.

Practice Questions:

Question 1
I roll a 21-sided die. How many outcomes are there?

Answer

Question 2
Give an example of an experiment that was not discussed above.

Answer

Question 3
In rolling a 6-sided die, I list seven outcomes: I roll a 1, I roll a 2, I roll a 3, I roll a 4, I roll a 5, I roll a 6, and I roll an even number. Are there actually seven outcomes?

Answer

Next Article: Some Basic Facts About Probability


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In our previous posts, we've talked about the basic concepts of probability and some fundamental facts about probabilities. Here, we'll show how to calculate the probability of a single event when all the outcome are equally likely. This is, in a sense, the simplest case that we will cover, and it is crucial for everything we'll do later (e.g. in finding the probability of two events occurring).

Suppose we flip a fair coin. What is the probability we get heads? Intuitively, the answer should be 1/2. And that's exactly what the following rule would say:

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.

So when we flip a fair coin, there are 2 possible outcomes (heads and tails). So n = 2 and the probability of one outcome (e.g. heads) occurring is 1/n = 1/2. And if we roll a six-sided die, there are 6 possible outcomes. So the probability of any particular outcome (e.g. rolling a 4) is 1/6. And if we held a raffle where there were 109 different entrants, the probability of any one of them winning would be 1/109.

Note that this rule only applies when all the outcomes are equally likely. In most GRE problems, the outcomes will be equally likely, and the question will signal that by saying that the outcome is "random" or that the outcomes are "equally likely." So, the question might say things like: "a name is chosen at random" or that "each outcome is equally likely." When the outcomes are not equally likely, all bets are off, and you will have to be more careful in how you approach the problem.

Now, we want to find the probability of some event occurring. Suppose I am going to roll a six-sided die, numbered 1 through 6. What is the probability that I get an even number? To calculate this, we use the following rule:

Probability of Single Events (for equally likely outcomes)
Suppose you have n equally likely outcomes. Then, the probability of some event E occurring is:

where the # of total outcomes = n.

So to find the probability of rolling an even number, we need to find the number of outcomes where we roll an even number. If we roll an even number, then we must have rolled a 2, 4, or 6. Then, we divide by the number of total outcomes, in our case 6. So, P(Roll an even number) = 3/6 = 1/2.

Here’s another example in a similar spirit:

Example 1
Suppose you randomly choose a number from 1 to 50. What is the probability that you chose a prime number?

Answer

Now we know how to find the probability of a single event when the possible outcomes are equally likely. Our next step is to learn how to combine these probabilities in order to get the probabilities of more complex events.

Practice Problems

Question 1
130 people line up to buy raffle tickets. Every 10th person who buys a ticket gets a teddy bear as a promotional item. What is the probability that a randomly chosen person from the line will receive a teddy bear?

Answer

Question 2
You have 50 friends. 12 of them have blue hair. You randomly pick one of your friends to invite to dinner tomorrow. What is the probability that you invite a person with blue hair?

Answer

Question 3
You still have 50 friends. 12 of them still have blue hair. What is the probability that you do not invite a person with blue hair?

Answer

Next Article: Probability for Two Events to Both Occur - P(A and B)


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