COUNTING COUNTING OUTCOMES & OUTCOMES & THEORETICAL THEORETICAL PROBABILITYPROBABILITY

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TREE DIAGRAMSTREE DIAGRAMS

You can use tree diagrams to You can use tree diagrams to display and count possible choices. display and count possible choices.

This option of counting possible This option of counting possible choices is used when the choices is used when the possibilities are limited. possibilities are limited.

ExampleExample

A school team sells caps in two colors (blue or A school team sells caps in two colors (blue or white), two sizes (child or adult), and two white), two sizes (child or adult), and two fabrics (cotton or polyester). Draw a tree fabrics (cotton or polyester). Draw a tree diagram to find the number of cap choices.diagram to find the number of cap choices.

Blue

White

Child

Adult

Child

Adult

Cotton

Polyester

Cotton

Polyester

Cotton

Polyester

Cotton

Polyester

When you use tree diagrams to decide how many choices there are, count up how many options you have in the right hand column.

= 8 choices

COUNTING PRINCIPLECOUNTING PRINCIPLE

Another way to count choices is to use the Another way to count choices is to use the counting principle.counting principle.

If there are If there are mm ways of making one choice, ways of making one choice, and and nn ways of making a second choice, ways of making a second choice, then there are then there are m(n) m(n) ways of making the ways of making the first choice followed by the second.first choice followed by the second.

This option is particularly useful when a This option is particularly useful when a tree diagram would be too large to draw.tree diagram would be too large to draw.

ExampleExample

How many two-letter monograms How many two-letter monograms are possible?are possible?

Since there are 26 letters in the Since there are 26 letters in the alphabet, there would be:alphabet, there would be:

First choiceFirst choice Second choiceSecond choice 26 choices26 choices 26 choices 26 choices

So, 26(26) = 676 possible monogramsSo, 26(26) = 676 possible monograms

Finding probability by Finding probability by counting outcomescounting outcomes

You can count outcomes to help you You can count outcomes to help you find the find the theoretical probabilitytheoretical probability of an of an event in which outcomes are equally event in which outcomes are equally likely.likely.

A A sample spacesample space is a list of all is a list of all possible outcomes. You can use a possible outcomes. You can use a tree diagram to find a sample space. tree diagram to find a sample space. Then you can calculate probability.Then you can calculate probability.

ExampleExample Use a tree diagram to find the Use a tree diagram to find the

sample space for tossing two sample space for tossing two coins. Then find the probability of coins. Then find the probability of tossing two tails.tossing two tails.Heads

Tails

HeadsTailsHeads

Tails

Answer: There are 4 possible outcomes, one of which is tossing two tails: 1/4

•You can also use the probability formula to solve this problem:

•P(event) = # of favorable outcomes = 1

# of possible outcomes 4