Probability
Probability
Objectives:
Apply fundamental counting principle
Compute permutations
Compute combinations
Fundamental Counting Principle
With repetition
Without repetition
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.
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
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
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
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
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
Practice Problems
Get ½ Sheet of Pad Paper (Crosswise)
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
Permutation With repetition
Without repetition
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.
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
)!(
!
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
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
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
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
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
Practice Problems
Get ½ Sheet of Pad Paper (Crosswise)
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?
Combination
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
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
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
Practice Problems
Get ½ Sheet of Pad Paper (Crosswise)
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?
More Problems
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.