Chapter 7: Probability

Lesson 3: Multiplication Counting Principles

Vocabulary:

• with replacement: items are replaced between events.ex: A bag of marbles contains 3 black, 4 blue, and 5 red. What is the probability of picking a blue and then another blue one if the first is replaced.

• without replacement: items are not replaced between events. ex: A bag of marbles contains 3 black, 4 blue, and 5 red. What is the probability of picking a blue and then another blue one if the first is not replaced.

Example 1:

• Pete Seria decided to offer a special on his famous pizza pies. He limited his special to cheese pizzas with or without pepperoni and with a choice of thin, thick, or stuffed crust. Pete wondered how many different versions of pizzas were possible? Illustrate this situation with a tree diagram.

Example 1, cont.

• What if Pete also decided to offer his pies in four different sizes: individual, small medium, and large. How many different possible choices were now available?

PIZZA

WITHPEPPERONI

THIN

THICK

STUFFED

W/OPEPPERONI

THIN

THICK

STUFFED

Example 2:

• Suppose you have a 6 question T / F test. How many arrangements of answers are possible?

____ ____ ____ ____ ____ ____

Theorem - Selections With Replacement:

• Let S be a set with (n) elements. Then there are nk possible arrangements of (k) elements from S with replacement.

• Example 3: Using Example 2, what is the probability of getting a perfect score by guessing?

Example 4:

• How many ways can you answer a matching test with 6 questions (assuming that each is used only once)?

_____ ______ ____ ____ ____ ____

Factorial:

• for integer n>0, n factorial

__________________________

Theorem - Selections Without Replacement: Let S be a set with (n) elements. Then there are

________ possible arrangements of the (n) elements _______________ replacement.

! ( 1) ( 2) ( 3)... 2 1n n n n n

n!without

Example 5:

Evaluate: a) 8! = b) 64! = c) =

87!

85!

Example 6:

• License plates in NJ have 3 letters (excluding I and O) and 3 numbers. How many combinations are there?

Closure

Example 7: All Mixed Up:There are 8 movie DVDs, 3 exercise DVDs, and 5 cartoon DVDs on the shelf. Suppose two DVDs are to be selected at random from the shelf. Find each probability.

a. P(selecting 2 movie DVDs) if no replacement occurs =

b. P(selecting 2 movie DVDs) if replacement occurs = c. P(selecting an exercise DVD, then a cartoon DVD) if no

replacement occurs =