COUNTING OUTCOMES
& THEORETICAL
PROBABILITY
12-4
TREE DIAGRAMS
You can use tree diagrams to display
and count possible choices.
 This option of counting possible choices
is used when the possibilities are
limited.

Example

A school team sells caps in two colors (blue or
white), two sizes (child or adult), and two fabrics
(cotton or polyester). Draw a tree diagram to find
the number of cap choices.Cotton
When you use tree
Child
diagrams to decide
Polyester
Blue
how many choices
Cotton
Adult
there are, count up
Polyester
how many options
Cotton
you have in the
Polyester
Child
right hand column.
White
Cotton
Adult
Polyester
= 8 choices
COUNTING PRINCIPLE


Another way to count choices is to use the
counting principle.
If there are m ways of making one choice,
and n ways of making a second choice, then
there are m(n) ways of making the first
choice followed by the second.
 This option is particularly useful when a tree
diagram would be too large to draw.
Example
How many two-letter monograms are
possible?
 Since there are 26 letters in the
alphabet, there would be:
 First choice
Second choice
26 choices
26 choices

So, 26(26) = 676 possible monograms
Finding probability by counting
outcomes
You can count outcomes to help you
find the theoretical probability of an
event in which outcomes are equally
likely.
 A sample space is a list of all possible
outcomes. You can use a tree diagram
to find a sample space. Then you can
calculate probability.

Example

Use a tree diagram to find the sample
space for tossing two coins. Then find
the probability of tossing two tails.
Heads
Heads
Tails
Heads
Tails
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