(1) The number of adults in City A is 1000.

Statement (1) is not sufficient to find the number of unemployed adults in City A.

(2) Three fourth of adults in City A are employed.

This statement tell us that of adults are unemployed. However, without knowing the number of adults in City A we cannot determine the number of unemployed adults in City A. Statement (2) alone is not sufficient.

Combining both statements, we find that the number of unemployed adults is of 1000 which is 250.

Both statements together are sufficient, but neither statement alone is sufficient.

(1) Half of the men in the sample do not have a beard.

This statement implies that the other half of the men in the sample have a beard. Therefore, the probability that a randomly selected man in the sample has a beard is 0.5.

Statement (1) alone is sufficient. 

(2) There are 100 men with a beard in the sample.

Statement (2) gives us the number of men who have a beard in the sample but without knowing the number of men in the sample, we cannot find the probability of men who have a beard.

Statement (2) is not sufficient.

(1) 1000 of the students at the university are graduate students.

Statement (1) alone is not sufficient because we don't know the total number of students enrolled at the university.

(2) Only 20% of the students enrolled are graduate students.

Statement (2) gives the probability of graduate students enrolled at that university as 20%(0.2).

Statement (2) Alone is sufficient.

In order to calculate the probability the question asks for, we need to know the number of each color of marble.

I) Gives us the number of greens.

II) Gives us clues which will allow us to find the number of reds and yellows.

We need both statements to answer this question.

To find the desired probability we need to know which color cards were removed.

I) Tells us the number of black cards removed.

II) Tells us the number of red cards removed.

At first glance, we need both statements, but if we think for a minute, if we know the total number of cards removed and the number of one color removed, we can find the number of the color... So either statement works.

The statements together do not provide a definitive answer. The question addresses theoretical probability, but the two statements together provide empirical results. While the two experiments strongly suggest that Holly removed mostly red cards, it is entirely possible for these results to happen with an evenly distributed deck, or even one which has all of its red cards left.

The probability of a single toss of a fair coin coming up heads (or tails) is ; the probability of five such outcomes in a row is .

Assume Statement 1 alone. Let  be the probability that a single toss of the coin will come up heads. The probability of five such outcomes in a row will be , which is greater than  by Statement 1. Therefore,

,

The probability of one toss of the coin coming up heads increased.

If Statement 2 alone is assumed, a similar argument shows that the probability of one toss of the coin coming up tails decreased - which, of course, is the equivalent outcome.

Statement 1 alone only gives the probability that any of three rolls, "1", "3", or "5", comes up, but it does not give us the probability of a "2" or a "4". The reverse holds for Statement 2. Neither alone gives us the complete picture.

From the two statements together, it can be deduced that the probability of the die not coming up a "6" is the sum , so the probability of the die coming up a "6" is . Since the probability of the die coming up "6" was  before it was altered, the alteration increased this probability.

From either statement, the length of one edge of a cube can be determined:

If the surface area is 150, then 

If the volume is 125, then 

Either way, since the length of each edge is known to be 5, the area of each of , , and  can be calculated by multiplying one half by the product of the legs - that is, one half by the square of 5. 

Also, since the three triangles are all right triangles with the same leg lengths, by the Side-Angle-Side Theorem, they are congruent, and their diagonals are as well - this makes  equilateral. Also, since each triangle is a right isosceles triangle, by the 45-45-90 Theorem, each hypotenuse can be calculated by multiplying 5 by . Therefore, the sidelength of  can be calculated, and its area can be determined. 

Therefore, each statement alone is enough to yield the area of each face - and the total surface area.

From Statement 1 alone, the probability of drawing a red number from the box before the addition of more marbles can be calculated - it is  - but no clue as to the number of red or white marbles in the box after the addition is provided. Statement 2 alone gives no clue about the numbers of marbles before or after the addition.

The two statements together provide sufficient information. If half the marbles in the box before the addition are red, as given in Statement 1, and half the marbles added are red - half the marbles are white, from Statement 2, and no other colors were added - then half the marbles in the box after the addition are red, and the probability remains  .