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Basic Counting Principle
The Basic Counting Principle
• When there are m ways to do one thing,
and n ways to do another,
then there are m×n ways of doing both.
• Example: you have 3 shirts and 4 pants.
– That means 3×4=12 different outfits.
• Example: There are 6 flavors of ice-cream,
and 3 different cones.
– That means 6×3=18 different single-scoop ice-creams
you could order.
When you have more than 2 choices:
There are 2 body styles:
sedan or hatchback
There are 5 colors available
There are 3 models:
How many total choices?
•GL (standard model),
•SS (sports model with
bigger engine)
•SL (luxury model with
leather seats)
Example 1
Sarah goes to her local pizza parlor and orders a pizza.
She can choose either a large or a medium pizza.
She has a choice of three different toppings(Shrimp, pineapple, sausage).
She can have two different choices of crust(thick and thin).
Predict how many different pizzas are possible using the counting princlple.
Draw a Tree Diagram.
Example 3
For her literature course, Rachel has to choose one novel to study from a
list of four, one poem from a list of six and one short story from a list of
five.
How many different choices does Rachel have?
Example 2
Derek must choose a four-digit PIN number.
Each digit can be chosen from 0 to 9.
How many different possible PIN numbers can Derek choose?
How many choices for first number(digit)?
How many choices for second number(digit)?
How many choices for third number(digit)?
How many choices for fourth number(digit)?
Write an expression for the total possible combinations.
More counting principle
• I want to generate a 3 letters password.
– The first letter must be a vowel.
– The second letter must be a consonant.
– The 3rd letter can be any letter
• How many choices for the 1st letter?
•
2nd letter?
•
3rd letter?
• Write an expression for possible passwords
Summary
Remember: The Counting Principle is easy!
Simply MULTIPLY the number of ways each activity can occur.
Basic Probability
• Number of favorable outcomes / total
outcomes
• You have 7 red balls and 4 blue balls
– What is the probability of drawing a red ball
Probability of drawing a vowel?
I have a box with all the letters in the English alphabet.
•What is the number of desirable outcomes?
•What is the number of total outcomes?
•What is the probability of drawing a vowel?
Probability (not)
A jar contains 9 black, 10 blue, 30 yellow, and 26 green
marbles. A marble is drawn at random.
What is the Probability of not drawing a Green?
• Number of desirable outcomes?
•Number of total possible outcomes?
•Calculated Probability?
P (not green)
Probability with Counting Principle
• You want to generate a random 4 password from
letters in the alphabet.
– What is the probability all the letters in the code will
be vowels?
• What are the number of desirable outcomes?
– How many vowels are in the alphabet?
– How many possibilities for 1st letter?
– How many possibilities for 2nd, 3rd, 4th letter?
• Write an expression for total desirable outcomes.
Probability with Counting Principle
(cont)
• What are the number of possible outcomes?
– How many letters are in the alphabet?
– How many possibilities for 1st letter?
– How many possibilities for 2nd, 3rd, 4th letter?
• Write an expression for total posasible
outcomes.
• Write an expression for the probability of all
vowels in the code.
Permutations
• ORDER IS IMPORTANT
– 1,2,3 is different from 1,3,2 and 2,1,3
• Factorial
– 7! = 7*6*5*4*3*2*1
• Cancelling Factorials
• Permutation Formula
Permutation Examples
• There are 8 students in a play. How many
ways can you arrange 6 students to come onto
the stage?
• You want to arrange the entire class (33
students) in a line. How many different ways
to you arrange the students?
More examples
• You want to elect a class leader and backup
leader from the class. How many different can
you make this selection?
Permutations comparisons
• Same sample size, but different number
chosen
• Larger sample size, but same number chosen
• Arrange 10 people in a line or arrange 9
people in a line