Since this problem deals with a probability with two potential outcomes, it is a binomial distribution, and so the probability of an event is given as:

Where  is the number of events,  is the number of "successes" (in this case, a "heads" outcome), and  is the probability of success (in this case, fifty percent).

One approach is to calculate the probability of flipping no heads, one head, two heads, etc., all the way to six heads, and adding those probabilities together, but that would be time consuming. Rather, calculate the probability of flipping seven heads. The complement to that would then be the sum of all other flip probabilities, which is what the problem calls for:

Therefore, the probability of six or fewer heads is:

Set A: 

Set B: 

In Set A, there are five consonants out of a total of seven letters, so the probability of drawing a consonant from Set A is .

In Set B, there are three consonants out of a total of six letters, so the probability of drawing a consonant from Set B is .

The question asks for the probability of drawing two consonants, meaning the probability of drawing a constant from Set A and Set B, so probability of the intersection of the two events is the product of the two probabilities:

Set A: 

Set B: 

In Set A, there are five consonants out of a total of seven letters, so the probability of drawing a consonant from Set A is .

In Set B, there are three consonants out of a total of six letters, so the probability of drawing a consonant from Set B is .

The question asks for the probability of drawing at least one consonant, which can be interpreted as a union of events. To calculate the probability of a union, sum the probability of each event and subtract the intersection:

The interesection is:

So, we can find the probability of drawing at least one consonant:

Set A: 

Set B: 

Between Set A and Set B, there are two potential matching pairs of letters: AA and XX. The amount of possible combinations is the number of values in Set A, multiplied by the number of values in Set B, .

Therefore, the probability of drawing a matching set is:

First calculate the number of students:

The probability of drawing an honors student will then be the total number of honors students divided by the total number of students attending the school:

First calculate the number of students:

The percentage of seniors that do not attend honors classes is:

Therefore, the probability of selecting a student who is a senior and one who does not attend honors classes is:

To calculate the probability of an event happening, take the number of ways to draw a green marble (5, because there are 5 possible green marbles to choose) by the total number of ways to draw a marble from the jar (19). Thus the answer is 

To calculate a probability, divide the possible ways to get the desire outcome by the total possible number of outcomes. We only care about rolling a 2, which can happen in 2 different ways (because there are two 2s on the die).

Thus we get:

To find the probability that two dice sum to seven we need to figure out the total number of ways that can happen. Represent the rolls as an ordered pair, with the first number in the pair corresponding to the first roll, and the second to the second roll. Then all the ways to get a sum of 7 are as follows:

 - or 6 different ways.

There are  possible outcomes when rolling two dice (6 different ways the first roll could come out, and 6 ways the second roll could come out)
thus the probability the sum is 7 is:

For a probability question like this, first do a sum of the total possible outcomes. For the data given, this is  or . Now, the probability of drawing a dish is . After this, there are  items in the box. The probability of drawing a spoon after this is .

Now, when two events are sequential like this, the probability of the two together is calculated by multiplying their respective probabilities. Thus, your total probability is  or