Probabilities can be found by the following equation:  , where the total outcomes is the sum of all of the different number of pets, and the numinator is the number of pets we want to pick (there are only  cats).  In our case, the probability is simply .

This is a combination problem. The solution to the question can be found by considering how many dogs can win the first prize (all five of them)! Then, if one has already won a prize, how many can win the second prize (four). This is because one of the five dogs has already been awarded a prize. Then, how many options are left to win the third prize? The answer is three. In a combination problem, we always multiply all of the possible outcomes for the results to get a total possible outcome value:  

Consider first all of the possible ways the men may be arranged, which is

Now, consider all of the ways that Aaron and Gary could be seated beside each other; it may be easier to visualize by drawing it out:

1. A G _ _
2. G A _ _
3. _ A G _
4. _ G A _
5. _ _ A G
6. _ _ G A

As seen, there are six possibilities.

Finally, for each of these cases, Craige and Boone could be seated in one of two ways.

So the probability that Aaron and Gary will be seated beside each other is:

243 applicants did not get a scholarship. This ratio is 9:243, which reduces to 1:27.

Probability is the likelihood that an outcome will occur, and it is calculated by dividing the number of favorable events by the total number of possible events that may occur. In order to answer the question, we need to divide the number of non-red and non-blue marbles (i.e. the number of favorable events) by the total number of marbles in the bag (i.e. the total number of possible events). The total number of non-red, non-blue marbles is the same as the number of yellow and green marbles

If the flowers that go in the first two locations are already determined, there are only three positions that run any risk of having the wrong flower planted in them. First, determine the number of possible ways in which Caroline can finish planting the garden. There are 3 possible flowers that she could plant in the third position. After she plants a flower there, there will be 2 flowers she can plant in the fourth position. After planting a flower in the fourth position, there will be only one flower left to plant in the fifth position; therefore, the number of possible ways she could finish the garden is:

There are 6 possible outcomes. Only one of these outcomes will be successful; thus, the probability that Caroline finishes planting the garden in the correct order is one out of six:

From rolling 2 dice, there are 36 possible outcomes (6 sides per die for both dice: 6 * 6 = 36). There are 4 possible outcomes that will result in a sum of 9 (3 and 6, 4 and 5, 5 and 4, 6 and 3). The probabilty of rolling a sum of 9 is  or 

When Jane starts she has a total of 30 pens, 9 black, 10 blue and 11 red. Two of the blue pens were lost. This dropped the number of blue pens to 8 and the total count of pens to 28.

To find the probability of something happening, take the total amount of things in question, the blue pens or 8, and compare them to total amount of pens which is now 28 pens.

 chance to draw a blue pen from the bag.

*The pitfall for these types of questions is to realize that not only does the count of blue pens decrease but also the total count of pens. Make sure to adjust both values. 

To find the probability of an event, find the number of ways that even could happen divided by the total number of outcomes of the event.

Number of ways we could spin a number divisible by 3: 
3,6,9 for 3 total
Number of different outcomes: 10.

Thus the probability is:

Probability of getting a head: 0.5
Probability of getting a tail: 0.5


The probability of not getting ANY tails = 0.5 * 0.5 * 0.5 * 0.5 * 0.5= 0.03125

The probability of getting AT LEAST 1 tail = 1 - 0.03125 = 0.969