A whole number is divisible by 6 if and only if it is divisible by all of the factors of 6, which, other than 1 and 6 itself, are 2 and 3. Therefore, it must be established that the number is divisible by both of these numbers. To be divisible by 2, it is necessary and sufficient that the last digit of the number be 0, 2, 4, 6, or 8. To be divisible by 3, it is necessary and sufficient that the sum of the digits be divisible by 3. 

Statement 1 alone, therefore, proves that the number is divisible by 2; however, it does not address divisibility by 3. Similarly, Statement 2 alone proves that the number is divisible by 3 (since 24 is as well), but it does not address divisibility by 2. The two statements together, however, address divisibility by 6, since if both are true, it holds that the number is divisible by both 2 and 3, and, consequently, by 6.

Assume Statement 1 alone. A necessary and sufficient condition for a whole number to be divisible by 5 is that its last digit be either 0 or 5. By Statement 1, the number meets this requirement, so it is divisible by 5. Statement 2, which gives the digit sum, is therefore, irrelevant.

From statement 1, we know the probabilty of choosing the first marble. However, since the marble is not replaced, it is impossible to calculate the probability of choosing the second marble. By knowing the information in statement 2 combined with statement 1, we can calculate the total number of marbles initially present.

Binomial Table

Statements A, B, C, and D are all true - so - 

The only false statement is E (the statement that declares that A and B and C and D are all false)

To determine the probability that the marble is yellow we need to know two things: the number of yellow marbles, and the number of marbles total. The first quantity divided by the last quantity is our probablility.

But the two given statements together only tell us that eighteen marbles are not yellow. This is not enough information. For example, if there are two yellow marbles, the probability of drawing a yellow marble is . But if there are twenty-two yellow marbles, the probability of drawing a yellow marble is  

Therefore, the answer is that both statements together are insufficient to answer the question. 

To answer this question you need to know two things: the number of red cards left and the number of total cards left. The second statement is irrelevant, as a reshuffle does not change the composition of the deck. The first statement tells you that there is one fewer red card than black cards, but it does not tell you how many of each there are, as you do not know how many decks of cards there were. 

And that information, which is not given, affects the answer. For example, if there were four decks, there were 103 red cards out of 207; if there were six decks, there were 155 red cards out of 311. The probabilities would be, respectively,

and 

, 

a small difference, but nonetheless, a difference.

The correct answer is that both statements together are insufficient to answer the question.

From statement 1, we learn that there are only two different colored marbles in the bag. From statement 2, we learn that  are black which tells us that are red. Without statement 1, it is impossible to determine if there is another color of marble in the bag.

standard deviation of the difference between the sample means = 

 at 95% (2-tailed) = 1.96

the sample difference must be 4 or greater

(note: the probability of the sample difference being -4 or less is so small (4.5 standard deviations) that it will be ignored and we will only consider the probability that the difference is 4 or more.) 

the probablity of the sample difference being 4 or greater (knowing that the population difference is 5) =

the table shows that .3156 lies below -.48, so, .6844 lies above -.48

In English - there is a .6844 probability that the 2 sample means will yield a sample difference that is 1.96 or more standard deviations above 0.   

575,757 must be divided by -