# Tutor profile: Mahati V.

## Questions

### Subject: SAT II Mathematics Level 2

Given a standard deck of cards, when you draw 4 cards without replacement, what is the probability of drawing at least 1 spade? Note that a standard deck of cards consists of 52 cards. There are 13 of each suit (spades, diamonds, hearts, clubs) and there are 4 of each value (A, 2, 3, 4, ..., J, Q, K).

This is a classic problem of complementary probability, and the key to identifying the problem as such is the phrase "at least." In this problem, in order to calculate the probability of drawing AT LEAST 1 spade, we can simply calculate the probability of drawing 0 spades, $$x$$, and take the complement of this probability ($$1-x$$). $$\bullet$$ So, the probability of NOT drawing a spade on the first draw is $$\frac{39}{52}$$ - because there are 13 spades and thus 39 non-spades in the deck. $$\bullet$$ We indicated that we are not replacing the cards upon drawing each card, so the probability of NOT drawing a spade on the second draw is $$\frac{38}{51}$$. $$\bullet$$ The probability of NOT drawing a spade on the third draw is thus $$\frac{37}{50}$$. $$\bullet$$ And finally, the probability of NOT drawing a spade on the fourth and final draw is $$\frac{36}{49}$$. Because each of these is an independent event, we multiply out these probabilities to obtain the probability of not drawing a spade on any of the 4 draws. $$\frac{39}{52} \cdot \frac{38}{51} \cdot \frac{37}{50} \cdot \frac{36}{49} = \frac{6327}{20825}$$ Of course, we need to ensure we take the complement of this probability to actually obtain the probability of drawing at least 1 spade in these 4 draws: $$1 - \frac{6327}{20825} = \frac{14498}{20825} \approx \fbox{0.696} $$

### Subject: Writing

What is the best way to plan out and structure an argumentative essay?

While there isn't a true right or wrong way to structure an argumentative essay, here are a couple good practices to follow: $$\bullet$$ Outline your essay before you start writing it. $$ \bullet$$ Start with a thesis statement (one sentence) that represents your take on the issue or topic at hand. Ensure that the thesis statement carries a level of complexity that will hint at your reader what the essay will be about. $$ \bullet$$ Then, generate topic sentences for each of your 2-4 body paragraphs. These topic sentences can be refined for transitions and flow later on when you flesh out your essay, but each topic sentence should represent a distinct idea that will be represented by the particular body paragraph. $$ \bullet$$ For each topic sentence, list out the evidence you will be citing to support each viewpoint. $$ \bullet$$ Within these 2-4 body paragraphs, include reasons supporting your thesis, but also include conflicting viewpoints or something that critics may bring up. Ensure that each conflicting viewpoint is negated with evidence-based reasoning supporting the thesis. $$\bullet$$ Your intro and conclusion should, more or less, be written last because the core of an argumentative essay is in the reasoning provided to back up your thesis statement. $$ \bullet$$ Intro: Ensure that you have some kind of attention-grabber to draw the reader in, and build up to the thesis statement (which should generally be the last sentence of the introduction paragraph). $$ \bullet$$ Conclusion: Keep it short and sweet. This shouldn't just be a rehash of the entire essay but it also shouldn't be the introduction of a new viewpoint. Connect back to your introduction, and give the reader something to think about. $$ \bullet$$ Other key aspects to keep in mind: $$ \bullet$$ Include strategic transitions! These keep your essay coherent and flowing as one, as opposed to just 5 or 6 disconnected paragraphs. $$ \bullet$$ Pay attention to your diction and syntax. As you go back and revise, ensure that every word serves a meaningful purpose. One strategically-selected and descriptive word carries a lot more meaning than a string of filler words. Vary your syntax. Long, winding sentences punctuated by a short, choppy sentence can help break up paragraphs into digestible thoughts. $$ \bullet$$ Cite your sources! The key to an effective argumentative essay is the supporting examples that lend ethos, or credibility, to the argument you are making. And that's it! These general tips have aided me in pretty much every kind of argumentative essay I've written, whether it's a timed AP English Language & Composition essay where I only have 40 minutes to respond to the prompt, a class assignment where I have had a month to write the essay, or even when putting my thoughts together for a legislative debate speech.

### Subject: Algebra

Bob's mom's age is 3 times Bob's current age. 5 years ago, Bob's mom's age was 4 times Bob's age at the time. How much older than Bob is Bob's mom?

For a short problem, there's definitely a lot to unpack in this problem! Let's start out by putting the words into equations. Today: $$\bullet$$ Let $$x$$ represent Bob's age today. We know that today, Bob's mom's age is 3 times Bob's age, so Bob's mom's age today is $$3x$$. 5 years ago: $$\bullet$$ Now, let's write Bob's and his mom's ages 5 years ago as a function of their current ages: $$ \bullet$$ 5 years ago, Bob was $$x-5$$ years old. $$ \bullet$$ 5 years ago, Bob's mom was $$3x-5$$ years old. $$ \bullet$$ Another fact we know is that Bob's mom's age 5 years ago ($$3x-5$$) was 4 times Bob's age at the time ($$x-5$$). $$ \bullet$$ If we write this out, we get the following equation: $$4(x-5) = 3x-5$$ $$ \bullet$$ Since we have our equation, all we need to do now is solve for $$x$$, calculate both Bob's and his mom's ages today, and find their difference to determine how much older than Bob his mom is. $$ \bullet$$ Start by expanding out the left side of the equation with the distributive property. Then, move all the $$x$$'s to one side and the constants to the other side to solve for $$x$$. $$4x-20 = 3x-5 \Rightarrow x-20 = -5 \Rightarrow x = 15 $$ $$ \bullet$$ We now know that Bob's age today ($$x$$) is 15 years old, which means that his mother is $$3x$$, or 45 years old. $$ \bullet$$ Take the difference between the ages to get a final answer of $$\fbox{30}$$ years. $$ \bullet$$ We can also verify that the age difference is the same 5 years ago, when Bob was 10 years old and his mother was 40 years old.

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