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: •GL (standard model),•SS (sports model with bigger engine)•SL (luxury model with leather seats)

How many total choices?

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 greenmarbles. A marble is drawn at random.

What is the Probability of not drawing a Green? P (not green) • Number of desirable outcomes?•Number of total possible outcomes?

•Calculated Probability?

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