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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?. Sample Space and Tree Diagrams. - PowerPoint PPT Presentation
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Holt Algebra 2
7-3 Multiplication Counting Principles
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)
FlavorPlain
Vanilla
FruitPeaches
Strawberries
Bananas
Raspberries
Blueberries
NutsAlmonds
Peanuts
Walnuts
Holt Algebra 2
7-3 Multiplication Counting Principles
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.
Holt Algebra 2
7-3 Multiplication Counting Principles
Example 1A Continued
numberof
flavorstimes
numberof fruits
numberof nutstimes equals
numberof choices
2 5 3 = 30
There are 30 parfait choices.
Holt Algebra 2
7-3 Multiplication Counting Principles
Holt Algebra 2
7-3 Multiplication Counting Principles
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 letter10 10 10 10 24 24 = 5,760,000
There are 5,760,000 possible passwords.
Holt Algebra 2
7-3 Multiplication Counting Principles
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
numberof plot choices
numberof
possible endings
=number
of adventures
6 4 5 = 120
There are 120 adventures.
Holt Algebra 2
7-3 Multiplication Counting Principles
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.
Holt Algebra 2
7-3 Multiplication Counting Principles
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.
Holt Algebra 2
7-3 Multiplication Counting Principles
Example 2• Show the sample
space for tossing one penny and rolling one die. (H = heads, T = tails)
Holt Algebra 2
7-3 Multiplication Counting Principles
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.]
Holt Algebra 2
7-3 Multiplication Counting Principles
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.
•
Holt Algebra 2
7-3 Multiplication Counting Principles
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.
Holt Algebra 2
7-3 Multiplication Counting Principles
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?
Holt Algebra 2
7-3 Multiplication Counting Principles
Selections with Replacement• Let S be a set with n
elements. Then there are possible arrangements of k elements from S with replacement.
k
n
Holt Algebra 2
7-3 Multiplication Counting Principles
Example 5• Sarah decides to rank the five colleges
she plans on applying to. How many rankings can she make?
Holt Algebra 2
7-3 Multiplication Counting PrinciplesExample 6
• In how many ways can a team of 12 people be ordered if captain always takes number 1 spot?
Holt Algebra 2
7-3 Multiplication Counting Principles
Selections without Replacement• Let S be a set with n elements. Then there
are n! possible arrangements of the n elements without replacement.
Holt Algebra 2
7-3 Multiplication Counting Principles