Example 1A: Using the Fundamental Counting Principle To make a yogurt parfait, you choose one flavor of yogurt, one fruit topping, and one nut topping. How many parfait choices are there? Yogurt Parfait (choose 1 of each) Flavor Plain Vanilla Fruit Peaches Strawberries Bananas Raspberries Blueberries Nuts Almonds Peanuts Walnuts

Example 1A Continued number of flavors number of fruits number of nuts number of choices times times equals 2  5  3 = 30 There are 30 parfait choices.

Example 1B: Using the Fundamental Counting Principle A password for a site consists of 4 digits followed by 2 letters. The letters A and Z are not used, and each digit or letter many be used more than once. How many unique passwords are possible? digit digit digit digit letter letter 10  10  10  10  24  24 = 5,760,000 There are 5,760,000 possible passwords.

number of starting points Check It Out! Example 1a A “make-your-own-adventure” story lets you choose 6 starting points, gives 4 plot choices, and then has 5 possible endings. How many adventures are there? number of starting points number of possible endings number of plot choices number of adventures   = 6  4  5 = 120 There are 120 adventures.

Check It Out! Example 1b A password is 4 letters followed by 1 digit. Uppercase letters (A) and lowercase letters (a) may be used and are considered different. How many passwords are possible? Since both upper and lower case letters can be used, there are 52 possible letter choices. letter letter letter letter number 52  52  52  52  10 = 73,116,160 There are 73,116,160 possible passwords.

Sample Space and Tree Diagrams When attempting to determine a sample space (the possible outcomes from an experiment), it is often helpful to draw a diagram which illustrates how to arrive at the answer. One such diagram is a tree diagram.

Sample Space and Tree Diagrams In addition to helping determine the number of outcomes in a sample space, the tree diagram can be used to determine the probability of individual outcomes within the sample space. The probability of any outcome in the sample space is the product (multiply) of all possibilities along the path that represents that outcome on the tree diagram.

Example 2 Show the sample space for tossing one penny and rolling one die.   (H = heads, T = tails)

Example 2 continued By following the different paths in the tree diagram, we can arrive at the sample space. Sample space: { H1, H2, H3, H4, H5, H6,   T1, T2, T3, T4, T5, T6 } The probability of each of these outcomes is 1/2 • 1/6 = 1/12 [The Counting Principle could also verify that this answer yields the correct number of outcomes:  2 • 6 = 12 outcomes.]

Example 3 A family has three children.  How many outcomes are in the sample space that indicates the sex of the children?  Assume that the probability of male (M) and the probability of female (F) are each 1/2.

Example 3 continued Sample space: { MMM MMF MFM MFF FMM FMF FFM FFF } There are 8 outcomes in the sample space. The probability of each outcome is 1/2 • 1/2 • 1/2 = 1/8.

Example 4 A quiz has 10 “True or False” questions. If you guess on each question, what is a probability of getting each question right?

Selections with Replacement Let S be a set with n elements. Then there are possible arrangements of k elements from S with replacement.

Example 5 Sarah decides to rank the five colleges she plans on applying to. How many rankings can she make?

Selections without Replacement Let S be a set with n elements. Then there are n! possible arrangements of the n elements without replacement.