The average of a set of numbers is the sum of its elements divided by the number of elements in the set. If Jim averaged 78 for the first three tests of the semester, the sum of his first three test scores must have been 78 3 = 234. If his average after a total of five tests is 84, the sum of his scores on those five tests must be 84 5 = 420. The sum of Jim’s last two test scores must be 420 – 234 = 186. This means that the least Jim could have scored on one of the last two tests is 86 (because the maximum possible score is 100).
These events are independent, so the probability of them both occurring is the product of their respective probabilities. The probability of pigs flying is .01, and the probability of the Red Sox not winning the World Series is
So the probability of both events occurring is .01 .9 = .009.
This is a combination problem because the order in which you put the sandwich together doesn’t matter. Recall that the equation that is used to calculate the combinations of a subgroup of size r from a group of n elements is:
To figure out how many different sandwiches can be made, you have to run the calculations for each aspect of the sandwich—bread, meat, and condiment—and then multiply the answers together. Since you can choose one of three types of bread, the combination is:
The meat can be chosen in 4C1 = 4 different ways, and the accessories can be chosen in 5C2 = 10 different ways. The total number of different sandwiches that can be created given the conditions is 3 4 10 = 120.
From the information in the question, you know that there are either 3, 4, 5, or 6 blue socks in the drawer. It is easy to find the probability of randomly selecting two blue socks in each of these situations, and from this information you can use trial and error to find the answer.
- If there are 3 blue socks in the drawer, then the probability of picking two blue socks is 3⁄6 2⁄5 = 6⁄30 = 1⁄5.
- If there are 4 blue socks in the drawer, then the probability of picking two blue socks is 4⁄6 3⁄5 = 12⁄30 = 2⁄5.
- If there are 5 blue socks in the drawer, then the probability of picking two blue socks is 5⁄6 4⁄5 = 20⁄30 = 2⁄3. This meets the conditions needed, so the drawer contains 1 black sock.
The set must contain four integers and their opposites. Thus, the sum of any integer and its opposite is zero. Since the mean is found by adding together all the elements of a set and dividing that answer by the number of elements, the mean for this set is: .
Using the formula for group questions: