probabilty

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Probability

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Page 1: Probabilty

Probability

Page 2: Probabilty

Objectives:

Apply fundamental counting principle

Compute permutations

Compute combinations

Page 3: Probabilty

Fundamental Counting Principle

With repetition

Without repetition

Page 4: Probabilty

Fundamental Counting Principle

Fundamental Counting Principle can be used to determine the number of possible outcomes when there are two or more characteristics.

Fundamental Counting Principle states that if an event has m possible outcomes and another independent event has n possible outcomes, then

there are m* n possible outcomes for the

two events together.

Page 5: Probabilty

Fundamental Counting Principle

Lets start with a simple example.

For a college interview, Robert has to choose what to wear from the following: 4 slacks, 3 shirts, 2 shoes and 5 ties. How many possible outfits does he have to choose from?

4*3*2*5 = 120 outfits

Page 6: Probabilty

Fundamental Counting Principle

At a restaurant at Cedar Point, you have the choice of 8 different entrees, 2 different salads, 12 different drinks, & 6 different desserts.How many different dinners (one choice of each) can you choose?

8*2*12*6 = 1152 different dinners

Page 7: Probabilty

Fundamental Counting Principle with Repetition

Ohio Licenses plates have 3 #’s followed by 3 letters.

A. How many different licenses plates are possible if digits and letters can be repeated?

10*10*10*26*26*26 = 17,576,000 different plates

Page 8: Probabilty

Fundamental Counting Principle without

RepetitionB. How many plates are possible if digits and numbers cannot be repeated?

10*9*8*26*25*24 = 11,232,000 plates

Page 9: Probabilty

Fundamental Counting Principle

How many different 7 digit phone numbers are possible if the 1st digit cannot be a 0 or 1?

8*10*10*10*10*10*10 = 8,000,000 different numbers

Page 10: Probabilty

Practice Problems

Get ½ Sheet of Pad Paper (Crosswise)

Page 11: Probabilty

1. Police use photographs of various facial features to help witnesses identify suspects. One basic identification kit contains 195 hairlines, 99 eyes and eyebrows, 89 noses, 105 mouths, and 74 chins and cheeks. The developer of the identification kit claims that it can produce billions of different faces. Is this claim correct?

2. Determine how many different license plates are possible in 2 digits followed by 4 letters if(a) digits and letters can be repeated(b) digits and letters cannot be repeated

Page 12: Probabilty

Permutation With repetition

Without repetition

Page 13: Probabilty

Permutations

A Permutation is an arrangement of items in a particular order.

Notice, ORDER MATTERS!

To find the number of Permutations of n items, we can use the Fundamental Counting Principle or factorial notation.

Page 14: Probabilty

Finding Permutations of n Objects Taken r at a Time

To find the number of Permutations of n items chosen r at a time, you can use the formula

. 0 where nrrn

nrpn

)!(

!

Page 15: Probabilty

Permutations of n Objects Taken r at a Time

Find the number of ways to arrange 6 items in groups of 4 at a time where order matters.

Example 1

3602!

720

)!46(

!646

p

Page 16: Probabilty

From a club of 24 members, a President, Vice President, Secretary, Treasurer and Historian are to be elected. In how many ways can the offices be filled?

480,100,520*21*22*23*24

)!524(

!24524

19!

24! p

Example 2

Permutations of n Objects Taken r at a Time

Page 17: Probabilty

Finding Permutations with Repetition

The number of distinguishable permutations of n objects where one object isrepeated q1 times, another is repeated q2 times, and so on is:

!... ! !

!

21 kqqq

n

Page 18: Probabilty

Find the number of distinguishable permutations of the letters in a) OHIO and b) MISSISSIPPI.

Example 1A

122

24

2!

4!

a) OHIO

Finding Permutations with Repetition

Page 19: Probabilty

Find the number of distinguishable permutations of the letters in a) OHIO and b) MISSISSIPPI.

Example 1B

3465022424

39916800

2! 4! 4!

11!

b) MISSISSIPPI

Finding Permutations with Repetition

Page 20: Probabilty

Practice Problems

Get ½ Sheet of Pad Paper (Crosswise)

Page 21: Probabilty

1. How many 3 letter words can we make with the letters in the word LOVE? 

2. What is the total number of possible 5-letter arrangements of the letters  w, h, i, t, e,  if each letter is used only once in each arrangement?  

Page 22: Probabilty

Combination

Page 23: Probabilty

Combination

A Combination is an arrangement of r objects, WITHOUT regard to ORDER and without repetition, selected from n distinct objects is called a combination of n objects taken r at a time. 

)!(!

!

rnr

nCrn

Page 24: Probabilty

Find the number of ways to take 4 people and place them in groups of 3 at a time where order does not matter. 

Example 1

46

24

)!34(!3

!434

C

Combination

Page 25: Probabilty

You are going to draw 4 cards from a standard deck of 52 cards. How many different 4 card hands are possible?

Example 2

725,27048! 4!

52!

)!452(!4

!52452

C

Combination

Page 26: Probabilty

Practice Problems

Get ½ Sheet of Pad Paper (Crosswise)

Page 27: Probabilty

1. In how many ways can you select a committee of 3 people from a group of 12 members?

2. A man has, in his pocket, a silver dollar, a half-dollar, a quarter, a dime, a nickel, and a penny. If he reaches into his pocket and pulls out three coins, how many different sums may he have?

Page 28: Probabilty

More Problems

Page 29: Probabilty

1. In how many ways can you list your 3 favourite desserts, in order, from a menu of 10?

2. A women has 4 blouses, 3 skirts, and 5 pairs of shoes.  Assuming the woman does not care what she looks like, how many different outfits can she wear?

3. Jack is the Chairman of a committee. In how many ways can a committee of 5 be chosen from 8 people given that Jack must be one of them?

4. In how many ways can you select a committee of 3 people from a group of 12 members? The committee members consist of a chairperson, treasurer and a secretary.