In order to solve this standard we need to understand two primary components: probabilities and the property of independent events. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

An intersection is the likelihood that one event will occur with another.

We can calculate that the probability of rolling a one on a six-sided die is:

Now, let's determine the probability of flipping a coin to reveal the "heads" side. A coin has two sides: heads or tails. The probability of rolling heads is as follows:

An intersection is the probability of two events occurring simultaneously; therefore, we need to multiply the probabilities.

Convert to a percentage.

Step 1: If we have 3 quarters and each quarter has two outcomes, then there are  total outcomes that can come out of all of the rolls of the quarters. 

Those rolls are:

HHH, HHT, HTH, THH, THT, TTH, TTT, HTT.

Step 2: Determine how many outcomes have no more than  heads....

We cannot count HHH because it has  heads. 

We also cannot count TTT because it has no heads at all.

Step 3: We had  options and we can't take  options..so we have a total of  options left. 

Step 1: Find the probability of getting a green marble...

The probability of a green marble is .

Step 2: Find the probability of getting a blue marble (without replacement)

When we have without replacement, we do not put the first marble we take out back into the sack. In this case, the total number of marbles in the sack is  less.

Probability of a blue marble is .

Step 3: To find the probability of both events happening at the same time, we need to multiply the probabilities in the first two steps.

So, 

Step 4: Simplify...

 is the answer. 

The number of ways to choose four people to sit at the first table without regard to the arrangement at either table is the number of combinations of four people from a set of eight; once the people who will sit at the first table are chosen, the people who will sit at the second are automatically chosen. Therefore, the correct number of such arrangements is .

Since

,

then, setting ,

,

the correct number of arrangements.

In a deck of 52 cards, one-fourth of the cards, or 13 cards, are spades. Also, there are four aces, one of which is a spade and three of which are not spades. It follows that the removal of the spades and the aces results in the removal of

cards, to leave a modified deck of 

cards.

None of the two red threes (the three of clubs, the three of diamonds) were removed, so the probability of a random draw of a red three is

.

Because the last test cannot be either math or writing, the selection of an order in which the tests are taken must be looked at as a sequence of events as follows:

First, the test to be taken last must be one of three tests: reading, science, or social studies;

Second, the remaining four tests must be arranged in a particular order. There are four ways to schedule the first test, three remaining ways to schedule the third, two for the third, and one for the fourth.

By the multiplication principle, this makes

possible schedules.

The mode of a data set is the value that occurs most frequently in the set. If two or more values tie for most frequently occurring value, then the set has multiple modes.

We show that the question of the modes of the data set cannot be answered with certainty as follows:

Suppose  assumes a value in the data set - for example, let . the data set is 

,

in which the most frequently occurring value is 1, which appears . This makes 1, , the mode.

Now, suppose  assumes a value not already in the data set; for example, let . The data set becomes

,

in which two values, 2 (the ) and 8 each occur twice.

Therefore, from the information given, the mode(s) cannot be determined with any certainty. 

The mode of a data set is the value that occurs most frequently in the set. If two values tie for most frequently occurring value, then the set has two modes - it is bimodal.

The value 16 already occurs twice in the data set. For the set to be bimodal, one of the following must happen:

Case 1: , which is not possible, since 0 is not considered prime or composite.

Case 2:  must be equal to one of the other four known values, 10, 18, 21, or 25. however,  is given to be prime - that is, to have only two factors, 1 and itself. Each of 10, 18, 21, and 25 is composite, having factors not equal to 1 or itself, so  cannot assume any of these values. 

Case 3:  must be equal to one of the other four known values, 10, 18, 21, or 25. Then  must be equal to half of the selected number:

Only in one case does it hold that  is a prime integer. 

The correct response is one - .

The mode of a data set is the value that occurs most frequently in the set. If two values tie for most frequently occurring value, then the set has two modes - it is bimodal.

The value 29 already occurs three times in the data set. For the set to be bimodal,  must be equal to one of the values that occurs once - 17, 21, 27, 35, or 37. Since it is given that  is prime - having only two factors, 1 and itself -  can only be either of 17 and 37, the other three values having other factors. 

The correct response is two.

The median of a set of nine data values - an odd number - is the value that appears in the middle when the values are ranked. For 8 to be the median, 8 must be in the middle - that is, four values must appear before 8, and four values must appear after 8.

In the given data set, four values - 4, 5, 5, 6 - are known to be appear before 8, so  must appear after 8. Since  is an integer from 1 to 10,  can only be 8, 9, or 10. This makes three the correct response.