Transcript Counting Principles and Tree Diagramsx

COUNTING PRINCIPLES AND
TREE DIAGRAMS
FUNDAMENTAL
COUNTING PRINCIPLE
The Fundamental Counting Principle
states that if there are x ways to
choose a first item and y ways to
choose a second item, then there are
x(y) ways to choose all items.
FOR EXAMPLE
A telephone company is assigned a new area code and can issue new 7digit phone numbers. All phone numbers are equally likely.
Find the number of possible 7-digit phone numbers
Use the Fundamental Counting Principle:
1st digit 2nd digit 3rd digit 4th digit 5th digit 6th digit 7th digit
?
?
?
?
?
?
?
10
10
10
10
10
10
10
There are 10 choices for each digit (0-9), so there are
10(10)(10)(10)(10)(10)(10) = 10,000,000 phone number options
YOU TRY
A telephone company is assigned a new area code and can issue new 7digit phone numbers. All phone numbers are equally likely.
Find the probability of a phone number that does not contain an 8.
First, use the fundamental counting principle to find the number of
phone numbers that do not contain an 8.
9(9)(9)(9)(9)(9)(9) = 4,782,969
P(no 8) = 4,782,969 = .478
10,000,000
USING A TREE DIAGRAM
The Fundamental Counting Principle tells you only the number of
outcomes in some experiments, not what the outcomes are. A tree
diagram is a way to show all possible outcomes.
For example:
A pizza place specializes in two types of crust, sesame and plain, and
sells five different toppings, onions, olives, ham, green peppers, and
mushrooms. Create a tree diagram to show the options assuming there
is only one type of crust with one topping.
TREE DIAGRAMS
Using the Fundamental Counting
Principle, we know that we
should have 10 options: 2 pizza
crusts, 5 toppings…2(5)=10
YOU TRY
You are going on a trip. You can pack 2 pairs of pants, 3 shirts, and 2
sweaters for your vacation. Use a tree diagram to show all outfit options
you can make if each outfit consists of a pair of pants, a shirt, and a
sweater.
There are 12 total outfits to choose from
2(3)(2) = 12
THE ADDITION COUNTING
PRINCIPLE
If one group contains x objects and a second group contains y objects,
and the groups have no objects in common, then there are x + y options.
For example
How many items can you choose from Bergen’s Deli menu?
Sandwiches
Salads
Soups
Turkey
Ham
Roast Beef
Reuben
Cobb Salad
Taco Salad
Grill Chicken Salad
Tomato
Chicken Noodle
French Onion
None of the lists contains identical items, so use the Addition
Counting Principle.
Total Choices = Sandwiches + Salads + Soups
T=4+3+3
There are 10 total items to choose from