• Necessary Assumptions vs. Sufficient Assumptions

    • Definition of an assumption: An unstated premise or condition that must be true for the conclusion to hold.
    • Importance of assumptions in arguments: Assumptions form the backbone of logical reasoning and argument construction.
    • Brief explanation of necessary and sufficient assumptions: These are two types of assumptions that serve different roles in arguments.

    Necessary Assumptions – Definition

    • Explanation of necessary assumptions: These are assumptions that must be true for the conclusion to be valid. Without them, the argument falls apart.
    • Key features of necessary assumptions: They are essential for the argument but alone may not be enough to guarantee the conclusion.

    Necessary Assumptions – Examples

    • Argument 1: All dogs bark. My pet, Rover, is a dog.
      • Necessary Assumption: Rover is capable of barking.
    • Argument 2: If it rains, the picnic will be canceled. The picnic was canceled.
      • Necessary Assumption: It rained.
    • Argument 3: Every employee who works overtime gets a bonus. Jane got a bonus.
      • Necessary Assumption: Jane worked overtime.
    • Argument 4: Vegetarians do not eat meat. Sam does not eat meat.
      • Necessary Assumption: Sam is a vegetarian.
    • Argument 5: All citizens must pay taxes. John pays taxes.
      • Necessary Assumption: John is a citizen.

    Necessary Assumptions – Discussion

    • Recap of necessary assumptions: Necessary assumptions are crucial for the validity of the argument.
    • Common scenarios where necessary assumptions are made: Everyday reasoning, hypothesis testing in research, legal arguments, etc.

    Sufficient Assumptions – Definition

    • Explanation of sufficient assumptions: These are conditions that, if met, guarantee the truth of the conclusion.
    • Key features of sufficient assumptions: They are enough to ensure the conclusion but aren’t always necessary for the conclusion to be true.

    Sufficient Assumptions – Examples

    • Argument 1: John studied hard for the math test. Therefore, John will pass the math test.
      • Conclusion: John will pass the math test.
      • Sufficient Assumption: Studying hard for the math test guarantees passing the test.
    • Argument 2: It snowed heavily last night. Therefore, school will be canceled today.
      • Conclusion: School will be canceled today.
      • Sufficient Assumption: Heavy snowfall the night before results in school cancellation.
    • Argument 3: Sarah eats a balanced diet and exercises regularly. Therefore, Sarah is healthy.
      • Conclusion: Sarah is healthy.
      • Sufficient Assumption: Regular exercise and a balanced diet ensure good health.
    • Argument 4: Our team won the final game of the season. Therefore, we will top the league.
      • Conclusion: We top the league.
      • Sufficient Assumption: Winning the final game of the season ensures topping the league.
    • Argument 5: I can save $500 a month. Therefore, I can buy a car next year.
      • Conclusion: I can buy a car next year.
      • Sufficient Assumption: Saving $500 a month is enough to buy a car next year.

    In each of these arguments, the premise leads to a conclusion. The sufficient assumption in each case is the unstated premise that the condition described in the premise leads directly and unerringly to the conclusion.

    Sufficient Assumptions – Discussion

    • Recap of sufficient assumptions: Sufficient assumptions provide enough conditions to ensure the conclusion.
    • Common scenarios where sufficient assumptions are made: Planning, risk analysis, scientific predictions, etc.

    Necessary vs. Sufficient Assumptions

    Necessary assumptions are premises that must be true for the conclusion to be valid. If these assumptions are false or not met, the argument would fall apart.

    Sufficient assumptions, on the other hand, if true, are enough to guarantee the conclusion, but they are not the only way the conclusion could be true.

    Consider the examples:

    Argument 1: John studied hard for the math test. Therefore, John will pass the math test.

    • Sufficient Assumption: Studying hard for the math test guarantees passing the test.
    • However, studying hard is not a necessary condition for passing. John might also pass if, for example, he is already highly proficient in math, he gets lucky in guessing some answers, or the test is easier than expected.

    Argument 2: It snowed heavily last night. Therefore, school will be canceled today.

    • Sufficient Assumption: Heavy snowfall the night before results in school cancellation.
    • However, heavy snowfall is not a necessary condition for a school cancellation. School could be canceled for other reasons, like a power outage, a water main break, or other emergencies.

    Argument 3: Sarah eats a balanced diet and exercises regularly. Therefore, Sarah is healthy.

    • Sufficient Assumption: Regular exercise and a balanced diet ensure good health.
    • However, a balanced diet and regular exercise aren’t necessary conditions for being healthy. Sarah could be healthy due to other factors such as genetics or perhaps she takes a medication that keeps her healthy.

    Argument 4: Our team won the final game of the season. Therefore, we will top the league.

    • Sufficient Assumption: Winning the final game of the season ensures topping the league.
    • However, winning the final game is not a necessary condition for topping the league. The team could also top the league if the other leading teams lost their final games.

    Argument 5: I can save $500 a month. Therefore, I can buy a car next year.

    • Sufficient Assumption: Saving $500 a month is enough to buy a car next year.
    • However, saving $500 a month isn’t a necessary condition for buying a car next year. The person could come into an inheritance, win the lottery, or earn a substantial raise at work.

    These are just a few examples to illustrate why the assumptions listed are sufficient but not necessary. There are often multiple paths to a given outcome (conclusion), and a sufficient assumption represents just one of those paths.

    Consider however the necessary assumption examples provided earlier: 

    • Argument 1: All dogs bark. My pet, Rover, is a dog.
      • Necessary Assumption: Rover is capable of barking.
        This assumption is necessary but not sufficient. Just because Rover is capable of barking (assuming this to be true), it doesn’t guarantee that Rover will bark.
    • Argument 2: If it rains, the picnic will be canceled. The picnic was canceled.
      • Necessary Assumption: It rained.
        The assumption that it rained is necessary but not sufficient. The picnic could have been canceled for a number of other reasons as well (e.g., a lack of attendees, food safety concerns, etc.).
    • Argument 3: Every employee who works overtime gets a bonus. Jane got a bonus.
      • Necessary Assumption: Jane worked overtime.
        The assumption that Jane worked overtime is necessary but not sufficient. Jane might have received a bonus for reasons other than working overtime, such as excellent performance or reaching a sales target.
    • Argument 4: Vegetarians do not eat meat. Sam does not eat meat.
      • Necessary Assumption: Sam is a vegetarian.
        This assumption is necessary but not sufficient. Sam might not eat meat for reasons other than being a vegetarian, like health issues or personal preference.
    • Argument 5: All citizens must pay taxes. John pays taxes.
      • Necessary Assumption: John is a citizen.
        The assumption that John is a citizen is necessary but not sufficient. John might be paying taxes for reasons other than being a citizen, such as owning property or conducting business in the country.

    In each of these cases, while the assumption is necessary for the conclusion to be valid, it is not sufficient to guarantee the conclusion because there could be other contributing factors or reasons.

  • This post is about GMAT preparation and a lot more. It’s a post about having the right mindset for learning new life skills. Adopting a positive mindset starts by acknowledging the challenges that are more than just intellectual.

    In this post I will:

    1. Explain why GMAT preparation is hard.
    2. Define a mindset that makes your GMAT preparation journey more tolerable.
    3. Give you four personality tests.

    The idea of the post originated a few months ago.

    Some background information first

    I always tell my Gmatbuddys the following gmat preparation guidelines:

    1. Read problems to understand, not to solve.
    2. Don’t obsess over metrics.
    3. Embrace uncertainty.
    4. Don’t compare yourself to others.
    5. Lose urgency and embrace the learning process.
    6. ‘Know’ a concept – don’t ‘know of’ a concept. The difference is vast.
    7. To know a concept or problem, learn how to teach it.

    Resistance to advice

    I am always amazed at how resistant to common sense advice my students are. I find that I must remind them on a regular basis. In fact, so often that I discovered that my students perceived me as too harsh or tough.

    “Me?” I thought?

    I am too empathetic.

    It came as a surprise therefore when this happened:

    A true story

    Some months ago a frustrated student of mine said “Ayham, please, some empathy”. We were solving a problem and I was exerting some pressure to move forward.

    I had always considered myself empathetic so the reaction surprised me.

    The student almost screamed at me.

    I wasn’t empathetic it turns out, or at least not as empathetic as I had thought.

    In fact, if you look closely, you will realize that my words betray some lack of empathy on my part.

    Observe:

    “I had always considered myself an empathetic person.”

    me

    There is no better definition for lack of empathy I think.

    If you think you are empathetic then, you’re probably not.

    That was personality test number one.

    Now this is not a post about empathy, but what I did to become more empathetic to create a better learning experience in Gmatbuddy.

    Back to GMAT preparation

    To maintain the effectiveness of Gmatbuddy, I decided to subject myself to the horrors of balancing life, work, annoying GMAT showoffs, and preparing for a long period of time.

    I wanted to be in the shoes of my students to ultimately create a better gmat preparation plan by discovering any false assumptions I had.

    Even though many Gmatbuddys have made significant improvements, I am always left with the feeling that I can create something even better.

    I couldn’t go through the gmat prep phase again as there is too much bias and knowledge about the experience that I have long internalized.

    My goal was to overcome something that was intellectually challenging, thought provoking, life changing and that also put a lot of pressure on my partners in crime.

    This is the How:

    The solution was to retake a challenging online course in probability theory that I had struggled to complete in the past.

    Here’s a summary of my attempts with reasons for failure.

    Attempt #1 – 2018

    Got too overconfident, fell behind, and could no longer catch up.

    Attempt #2 – 2019

    Humbled by my previous experience, I took the second attempt seriously. But summer in Barcelona happened, family visits, and then you know, the rest was history.

    Attempt #3 – 2020 + Covid lockdown

    This time, I was going to do it. No compromise I thought:

    I paid for a verified track and told everyone in my family I was doing the course.

    Subjecting oneself to the pressure and the prospect of humiliation does wonders to the motivation.

    Lock down silver lining

    Now you might think COVID-19 came in the way to finishing the course but in fact, it gave me more time to work.

    I enjoyed the lockdown really. It has been about 10 weeks now (or a year?) and my sanity is in check (I think).

    Did you hate the lockdown?

    If Yes: You’re an extrovert.

    If No: You’re an introvert.

    That was personality test number 2.

    I really put a lot of time and effort into the course and discovered some incredible insights about probability.

    Some mind blowing things I tell you. Anyways, I will write a post about that later.

    Here however was another incredible discovery:

    I broke every single guideline I demand of my students.

    Every, Single, One.

    Here the GMAT preparation guidelines again:

    1. Read problems to understand, not to solve.
    2. Don’t obsess over metrics.
    3. Embrace uncertainty.
    4. Don’t compare yourself to others.
    5. Lose urgency and embrace the learning process.
    6. ‘Know’ a concept – don’t ‘know of’ a concept. The difference is vast.
    7. To know a concept or problem, learn how to teach it.

    Human nature is too powerful though, especially when the pursuit becomes about your ego.

    Gmat preparation tips
    Hey you – you’re fabulous

    Managing your inner fears is perhaps as challenging, if not more challenging than the actual GMAT, or any endeavor you pursue.

    I found myself breaking every single rule when I did the midterms, solved the problems sets, watched the lectures. I wanted more to finish the course than to learn the content.

    My mindset was to finish and see the ‘green tick’ of complete.

    I once laughed when a learner complained about the green tick not showing when they complete a task, but I found myself extremely bothered when it happened to me.

    Every time I complete a task I check my progress bar to see whether I am close to passing the class.

    It’s a tremendous urge that was hard to overcome.

    Every time I looked at a problem, I thought about how I can solve it instead of spending time to understand it in my own words.

    Not comparing myself to others was extremely difficult.

    It was especially hard when the occasional a@@$o@e goes on the forum

    and starts their post like this

    “So I scored 95% on the midterm but I have a doubt, why does….”

    First of all, Silentmather901, why do you

    1. Need to tell me you did well?

    2. Need to be validated, incognito mode, by strangers? It’s absurd.

    “What a loser I thought.”

    But then again, why was I so angry? I mean I don’t know who SilentMather901 is.

    Every gmat test taker can relate to hidden users showing off their performance on forums.

    It was just strange that I broke all my rules. 

    It was becoming about ego, not learning probability.

    I better understood the struggle that my students go through on the months they prepare, and the tremendous pressure it places on their families as well.

    That’s one thing I learned:

    If your family is not in on it, you’re going to have a hard time. Because it can be unfair, especially when you’re not seeing results and you go into the dark place.

    Imagine being seen in your ‘pyjamas’ at 2 am working on math problems looking like a lunatic.

    It’s not good. Don’t make it about ego and go down the rabbit hole.

    Remember,

    It says nothing about you if you have to spend more time than you think.

    It’s silly to compare yourself to others because we all have different experiences.

    Focus on yourself.

    I took it for granted that it’s easy to follow the common sense rules, but I was humbled by my experience.

    Once you’ve mastered a subject, it’s easy to forget that it’s hard to learn the basics and internalize them. Until then, don’t expect others to just switch them on.

    So I have become more empathetic. (I think).

    Note: if you’re wondering whether I passed the probability course then you have an ego problem.

    That’s personality test number 3.

    So, here are the GMAT preparation rules again:

    1. Read problems to understand, not to solve.
    2. Don’t obsess over metrics.
    3. Embrace uncertainty.
    4. Don’t compare yourself to others.
    5. Lose urgency and embrace the learning process.
    6. ‘Know’ a concept – don’t ‘know of’ a concept. The difference is vast.
    7. To know a concept or problem, learn how to teach it.

    It’s ok to forget.

    Remember to remember them.

    But it’s ok, no pressure if you break them sometimes.

    Ps: if you kept going to the end for personality test #4 then you’re obsessive compulsive.

    That’s personality test number 4.

    The end.

  • Two points to understand why you failed the GMAT

    To understand why you failed the GMAT, you must understand two things very well:

    • Mastery
    • What the GMAT is and what it measures

    Mastery

    Mastery is a term popularized by Malcolm Gladwell, and very briefly suggests that you must practice a skill for about 10000 hours to master that skill. You ‘master’ a skill when you are able to perform it with ease and fluidity. The 10000 hours of practice allow you to internalize the skill and hence achieve that fluidity. You can see where I am going with this. You failed the GMAT because you have not mastered certain skills required to do well on the test.

    What the GMAT is and what it measures

    The GMAT is test of reasoning.
    What is reasoning? the exercise of understanding problems, making informed and directed incremental inference points and connecting those inference points, to arrive at a final conclusion that solves the problem.

    Reasoning requires making inference, and making inference requires mastery of basic math and reading comprehension. Because every problem requires more than one inference point, you have to be able to iterate the reasoning cycle to understand and infer quickly and consistently.
    Because you have to connect those inference points, you must have vision and direction to move towards the final solution. So, if your foundations are weak, you will be slow, instead of solving a problem in 2 min, you need 3 min, instead of 1.5 min, 2 minutes.

    If your vision or direction is inexperienced, you will not see the next stage of the problem or even understand what the objective needs to be.

    Why you failed the GMAT

    And that is exactly why you have failed the GMAT – your foundations are not strong enough to reason good enough and fast enough towards a clear goal.
    You have not strengthen your foundations or quality of your reasoning and just redid problems and checked them of a list. You have to substantially improve your foundations and quality of your reasoning – this requires a strong hard look at how you understand and solve problems.
    That is exactly what the GMAT measure:

    the quality, consistency, speed and direction of your reasoning.

    So, unless you improve your foundations in basic math and reading comprehension, you will not improve the quality and speed of your reasoning. Unless you improve the direction and vision of your reasoning, you will not improve your reasoning, and you will not therefore, improve your score.

    How do you improve your GMAT score?

    Improve foundations

    basics, many many times – more time than you expect. Question, if I ask to solve this question, how long does it take you to solve it? When was the last time you read a book?

    Improve your reasoning

    A symptom of failure to reason is asking

    “how do I solve this problem” instead of “how do I understand this problem better”

    Hence, better reasoning starts with the desire to understand problems more than solve problems. When you are solving problems, your primary goal should be to understand the problem, more than obsessing about ‘how to solve the problem’.

    It’s a KEY distinction.

    If you clearly state the objectives of the problem, for example, x = ?, your direction and focus on understanding the problem in your own words, and the solution will follow naturally.

    Doing this consistently and improving your process, will improve your direction and vision in reasoning to understand better and ultimately solve problems.

    How long will it take?

    So let’s understand your score therefore as a measure of reasoning, knowing that your goal is to arrive at mastery in reasoning. For practical purposes, we can say that mastery is around the 700 mark. How long does it take to arrive? The further away you are from that score, the longer you have to prepare. If you score around the 400-500 range, you will need about 18 months of preparation, and there is no way around it.failed the GMAT

    How do some people do so well with little preparation then?

    Because they have already achieved mastery in basic math and reading comprehension before they even begin to prepare for the GMAT. They have already achieved mastery in reasoning before they even take a GMAT course. An engineer, lawyer, scientist, consultant, or any job function that demanded solving problems, and therefore reasoning, on a regular basis for a long time. They were already ready for the GMAT before they even new of the GMAT.

    You would have achieved mastery in foundations during high school if you had done your homework or were a diligent student.
    You would have achieved mastery in reasoning during university or your job if you had to solve problems on a regular basis for a long time.

    Accept the time that it will take

    Now obviously you don’t have to practice for 10000 hours because you have already spent time practicing basic math, reading comprehension and reasoning. Have you practiced for 8000 hours? 9000 hours? That is reflected in your score. Time to catch up.

    Understand who your up against : Someone who practiced their foundations for a long time and reasoned for a long time. Are you going to catch up in 2 months? 3 months? 4 months? You can’t and you must accept that. You have to put in the work.

    A psychological stand point

    The five stages of grief describe the phases we go through during a crisis:

    Denial, anger, bargaining, depression and acceptance.

    Understand where you are in that cycle to avoid endless frustration to create a realistic plan. Doing yet another GMAT prep course that promises 200 points in 2 months is not going to cut it because you have not mastered foundations of basic math and reading comprehension, and you have not mastered reasoning. If you want to improve your score, you must improve these foundations and you must improve your reasoning.

    Also remember that you don’t need to do this alone. Hire a tutor. He or she can point you in the right direction and help you make the most out of your time and effort by focusing on what matters.

  • Gmat anxiety is a main cause of test day under performance. It is common to tell oneself to ‘be less stressed’ or to ‘get over your anxiety’. But, there is little value in forcing yourself to calm down.

    Over the course of millions of years the human brain evolved into a complex structure capable of high order thinking. But there is still a building block of the brain, deep at its core, that is primitive and reactionary that is called the ‘lizard brain’. Naturally reactionary because it was necessary for our survival: to dodge a saber tooth tiger or duck as a rival tribe member takes a swing at you with their club. We had just enough time to react but not to think.

    This lizard brain still ‘exists’, and kicks in whenever we feel threatened. So, you can think of ‘gmat anxiety’ as a defense mechanism. Hardwired into our brain, it is almost impossible to consciously dominate. That is why we are unable to reason our way out of anxiousness. It is a survival mechanism. This is important to remember because on the test, there is no use telling yourself ‘not to be anxious’. The lizard brain is still at work. Sensing the threat that you’re running out of time, it will sabotage your common sense and coerce you into looking for answers. Of course, it is trying to help you, but, the lizard brain cannot see the big picture and does not have any common sense. It wants to save you and will therefore try to push you into choosing the right answer fast. That can be a good thing, but more often it is not. Especially when you still have not understood a wordy quant problem or a complex critical reasoning passage. Your first objective should be to understand problems. This is hard when you’re running out of time and the natural reflex is to look for answers instead of to understand problems.

    You should not fight gmat anxiety because you can’t. Instead, learn to accept your anxiousness and work with it. Visualize your test many times during the days and weeks leading up to the test. Brainwash your brain into a procedure that will kick in once your anxiousness kicks in. Learn to accept your anxiousness and when you do, you will be more likely to remember the most important thing you have to do on the test: Understand the problem.

    The lizard brain will try and force you to rush, look for answers and think about formulas before taking your time to think. Back in the day, there was no need to ponder because the problem was clear: survival. Now the problems aren’t always very clear and you must spend time to understand what they are first.

  • Logical thinking on the GMAT brings great returns. How good is your logical thinking?

    Situation: In a deck of cards, any card has a letter on one side and a number on the other. The rule is that if a card has X one side then it has a 2 on the other side.

    Question: What is the minimum number of cards you must turn to verify that rule?

    The answer choices are:

    A. Card X only

    B. Cards X and 2

    C. Cards X and 5

    D. Cards X, 2, and 5

    E. All cards

    The answer is:

    C. Cards X and 5.

    This may seem counterintuitive but in fact is logically the correct answer.

    A common response is B: Cards X and 2.

    Remember however that the rule is that ‘if X then 2’, and not ‘if 2 then X’. An equivalent but more familiar argument is the following: if I am in Barcelona then I am in Europe. However, that does not mean that if I am in Europe I am in Barcelona. From this, we can conclude that turning card X is indeed necessary to prove the rule, but not turning card 2.

    The trickier part is why we have to turn the card with the number 5 on it. Note that the question asks us for the movements we need to make to prove the statement ‘if X then 2’ given that every card has a letter on one side and a number on the other. What if we turn card 5 and find the letter X on it? Then the rule ‘if X then 2’ is false.

    Hence, we must turn the card with number 5 on it to prove the rule.

    This question is based on the Wason Selection test that is designed to test your logical thinking. The relevance to logical thinking on the GMAT is obvious. On a very abstract level, the Quant section tests logical thinking. The critical reasoning section is also a test of logical thinking, especially analysis of arguments and assumption questions. The good news is that logic is learnable skill. Remember to turn it on and improve it.