# Tutor profile: Andrew L.

## Questions

### Subject: SAT II Mathematics Level 2

Two fair six-sided dice are rolled. What is the probability that their product is greater than 3?

Unfortunately, this is a bit of a tedious question that may require a lot of scratching on scratch paper, so the key here is to try and solve it quickly and accurately. Remember that problems involving dice are generally not going to be horrendously complicated, so work quickly and confidently, knowing that the calculations will not blow up in your face. There are 36 possible combinations for the result of rolling two dice (given by 6 * 6 because there are six sides on a die), and a quick mental check reveals that most dice pairings will have a product greater than 3. So let's count the complement! We will find the probability that the product of the two dice is less than or equal to five, and then do 1 minus that quantity. How many combinations of dice result in a product less than or equal to 3? This is something we can do quickly. We realize at least one of the dice have to be a 1, because if neither dice was a 1, the smallest product possible is 2 * 2 = 4 > 3. So our combinations are (1, 1), (1, 2), (1, 3), (2, 1), and (3, 1). That's five combinations out of the total of 36, so the probability that the product of the two dice is less than or equal to 3 is 5/36. This means that the probability that the product of the two dice is greater than 3 is 1 - 5/36 = 31/36. Two things to be careful of here -- the opposite of "greater than 3" is "less than or equal to 3," not just "less than 3." And also, do not forget to do the final subtraction at the end! Remember to always answer the question that is asked.

### Subject: SAT

If 4x - y = 4, then what is the value of 16^x / 2^y? Express your answer in simplest form or specify that not enough information is given.

This problem is terrifying because of combination of unknowns with more unknowns with exponents and fractions, and also because of the possibility of there not being enough information to answer the question. But we will see that the problem is actually not too complex. We start with the given equation that 4x - y = 4. Since the expression whose value we are trying to find involves a relationship of x and y, it seems a solid idea to express x and y in terms of each other using the given equation. By a simple solving for y, we obtain that y = 4x - 4. We can then plug this value of y in to the expression we which to find the value of, and we get 16^x / 2^(4x - 4). Now noticing that 16 is a power of 2, we pull off a crucial move -- by leveraging the property of exponents, we can simplify 2^(4x - 4) = 2^(4 * (x - 1)) = (2^4)^(x - 1) = 16^(x - 1). Now, our expression is much simpler! 16^x / 16^(x - 1) = 16, again by the property of exponents. We can also see this conceptually by noting that 16^x = 16 * 16^(x - 1), and clearly 16 * 16^(x - 1) / 16^(x - 1) = 16.

### Subject: Algebra

Find the distance between the points (-2, 3) and (10, 8).

Many students may quickly apply the Distance Formula here, but I want to ensure that the student has a thorough understanding of the concept behind the Distance Formula. The distance between two points is the length of the direct line segment between them, and we can find the length of that segment by using an application of the Pythagorean Theorem. Since the x-axis and y-axis are perpendicular, we can draw (or imagine) a right triangle with the distance between the x-coordinates of the two points as the first leg, the distance between the y-coordinates of the two points as the second leg, and the direct line segment between the two points as the hypotenuse. Thus, we can see that the square of the distance between the two points is equal to the sum of the squares of (10 - (-2)) = 12 and (8 - 3) = 5. 12^2 + 5^2 = 169, so our distance comes out to a nice round 13 units.

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