Probability and the Binomial Distribution

Some helpful key concepts

Probability problems describe a random experiment and some specific event its probability you are asked to calculate.
The probability of the event is a fraction, a positive fraction between 0 and 1. This fraction is a smaller number divided by a bigger number.
The big number at the bottom of the fraction is the total number of all possible outcomes of the experiment, everything that can possibly happen in the experiment.
The small number at the top of the fraction is the count of exactly how many ways the very specific event we are interested in can come to happen.
Usually two different counting techniques are needed, one for calculating the big count, the denominator; and another for finding the smaller count, the numerator.
It is a good idea for GMAT or GRE test takers to learn well a formula called “the probability distribution of a binomial random variable.”
I know the name is quite a mouthful but go by this example:
A 60% heads, 40% tails biased coin is tossed 20 times.
The total number of times the coin lands “heads” is represented by X, and is called “the random variable” in this experiment.
The “number of trials” is represented by n, and it equals 20 in the above example.
The “probability of success” at each individual trial is represented by p, and it equals 60% in this example.
The “probability of failure” at each individual trial is represented by q, and it equals 40% in this example.
There is an established formula for calculating the probability that the number X will equal each integer value from 0 to n (0 to 20 in this example). You want to memorize said formula and get enough practice applying it until you feel quite comfortable using it in various situations.

Repetition Is Key

Practice makes perfect

Most students need several exposures to a procedure before being able to perform it correctly. Successful performance of mathematical procedures has several ingredients: understanding the concept is fundamental but is not enough. Memory plays an important part as well, you have to put in the effort required to commit the formulas to memory.

Then there are the concrete examples that give the abstract principles a solid meaning. The most important part is application to complex examples, how to actually use the formulas, how to recognize you have an expression in front of you that is calling for that particular formula. More often than not, each of these performance components needs the student to be exposed multiple times to the subject matter in order for the skill to be correctly installed into the student's mind. In these regard our brain differs vastly from computers. A typical computer program gets completely installed in one installation session. The programs our brain uses to solve mathematical problems generally need multiple installation sessions to become functional. Sometimes there seems to be a big, tall skepticism barrier guarding the mind of the student against any mathematical learning. Sometimes, with some students, I end up under the impression that I have to show them how the formulas work and what the connections are from every conceivable angle before they finally make the decision to believe in the process, to see it for themselves, and to use the tools. In these cases I feel they make every mistake they can possibly make, at every chance they get. I do not attribute this behavior to lack of intelligence but to a strong skepticism, some sort of lack of faith, and a need for attention. That is another reason repetition is so important, not only because it is required for most students to understand a procedure well, but also because some students need the teacher to have multiple opportunities to figure out a way to finally explain the subject to that particular student well enough that finally he or she makes the decision to trust the evidence that has been presented to him or her and adopts the procedure.
Practice makes perfect, for the student as well as for the teacher.

How to Make the Most of Private Tutoring

Useful tips for students looking for tutoring

Find a math tutor who is right for you.

Plan ahead, work out a budget, and find a tutor within your price range.

Make sure you feel comfortable communicating with your tutor by e-mail, phone, and in person.

Make sure you feel comfortable about the tutoring schedule, and the meeting place you use for the tutoring sessions, whether it is your own house, a coffee shop, a public library, or the tutor’s office.

Schedule at least one tutoring session per week through your exam’s date. Two weekly sessions 3 or 4 days apart from each other is the ideal schedule. Three weekly sessions may be necessary in some cases. One-hour-long sessions work best in most cases, although some students prefer 90-minute sessions. Your tutor should be flexible and accommodating about this. He or she should work with you to find a good working schedule. Once the schedule is set, strictly adhere to it as best you can.

Make sure the tutor clearly explains the subject matter, and answers your questions in a way that is easy for you to understand.

Bring all your study materials to the tutoring sessions: prep book(s), notebook, pencils/pens, scratch paper, and the same calculator you will use in the test (if one is allowed).

Set enough time apart to work out practice problems on your own before each tutoring session.

While you are working out practice problems by yourself, identify all the problems that give you trouble. Mark them with sticky notes, color flags, highlighter, or in any other way that reminds you what kind of trouble you are having with each particular problem.
Mark the problems you did not even have a clue how to approach them; those where you got the wrong answer; those that took you too long; those that you got right but you are not really sure why; those where you read the explanation provided in the book but you did not understand such explanation. Mark all such problems.

The purpose of the preparation work described above –prior to the actual tutoring– is for you to enable the tutor to maximize results in your benefit. You want your questions answered. You don’t want to pay money just so someone sits next to you watching you solving a bunch of problems that you can do on your own without any trouble.

During each tutoring session make sure to ask all questions that come to your mind. Do not allow you to keep the slightest bit of confusion to yourself, share it with the tutor. Help your tutor understand what is confusing you. Tell him or her how you feel, and what thoughts and/or emotions go through your mind when you read each problem. It is very important your tutor understands your thought processes (whatever they may be) so he or she can help you by pointing you in the right direction. When I say “the right direction” I mean the right direction for you with that particular problem. There are many problem-solving techniques and strategies available, and each student is uniquely suited to use some better than others. That is why telling your tutor all your reactions to a problem is so important.

Make notes about everything your tutor says that you find useful, new, insightful, or otherwise helpful in any way. Any problem-solving methods, techniques, strategies, rules, formulas, tricks or shortcuts your tutor shows to you, and you feel they will help you solve the problems faster and/or with greater accuracy or reliability, make a note of them to review them later, and apply them to similar problems.