Counting & Probability
Chapter 12
ckendall
1/22/2013
Honors Pre-Calculus
Chapter 12 Assignments
1/21 M: NO SCHOOL
T:
p. 907 1-37 eoo skip 21
W: p. 917 1, 12, 19, 23, 25-59 o
R: p. 929 7-15 o, 23-31 o
F:
p. 930 35,37-73 eoo
1/28 M: Review
T:
Review
W: Ch 12 Test
Honors Pre-Calculus
12.1
Sets & Counting
Learning Targets: Students will be able to analyze and interpret sets and apply counting theorems.
SET – a well defined collection of distinct objects (ELEMENTS)
 “well-defined” means there is a rule determining whether an object is an element of the set
 if a set has not elements, it is call the EMPTY SET, or null set,
 sets are denoted by capital letters; elements of a set are written in braces { }
Given two sets A and B:
 if A and B have exactly the same elements, then they are equal sets,
 if each element of set A is also an element of set B, then A is a subset of B
AB
A B
A B

if A  B and A  B , then A is a proper subset of B


if x is an element of set A, the notation is
the empty set is a subset of EVERY set,




the intersection of A and B, A  B , is the set consisting of elements of both A and B
if two sets have no elements is common, they are said to be disjoint
A B  
the union of A with B, A  B , is the set consisting of elements of found either A or B or BOTH
the universal set, U is the set consisting of all the elements in consideration
the set of real numbers R

the complement of a set, A , is the set consisting of all elements in the universal set that are not in A
sometimes denoted as A '
Examples
Given:
A = {1,3,5,7,9}
Given:
U = {0,1,2,3,4,5,6,7,8,9}
C = {1,2,4,6,8,9}
7.
A = {1,3,4,5,9}
AB
15.
  A, for any set A
B = {1,5,6,7}
 AC B C
6.
x A
 A  B  C
B = {2,4,6,7,8}
18.
C = {1,3,4,6}
B C
Counting
 if the number of elements in a set is a nonnegative integer, then the set is finite (otherwise it is infinite)
 the notation for the number of element in a set is n(A)

if A is a set with n elements, then A has 2 n subsets
Theorem:
Counting Formula
If A and B are finite sets, then
Theorem:
Addition Principle of Counting
If two sets A and B are disjoint, then
n  A  B   n  A  n  B   n  A  B 
n  A  B   n  A  n  B 
Examples
26.
If n  A  B   60, n  A  B   40, and n A   n B  , then find n  A
Use the Venn diagram on the right.
30.
How many are in A and B?
U
A
B
3
15
10
2
34.
How many are in A or B or C?
5
2
15
C
4
Example
Set up a Venn diagram to describe the following situation and answer the questions.
In a class of 33 students, 17 of the like Rocky Road ice cream, 16 like Mint Chocolate Chip, and 11 like plain Vanilla. 7
like both Rocky Road and Mint Chocolate Chip, 5 like both Rocky Road and Vanilla, and 3 like Mint Chocolate Chip and
Vanilla. Only 2 students like all three types of ice cream.
a.
How many like only Rocky Road?
b.
How many like either Vanilla or Rocky Road, and not Mint Chocolate Chip?
c.
How many didn’t like any of these flavors?
Honors Pre-Calculus
12.2
Permutations & Combinations
Learning Targets: Students will be able to differentiate and use permutations, combinations, and the multiplication
principle of counting.
Example:
If you can choose:
one of 3 kinds of ice cream (Vanilla, Chocolate, or Strawberry)
one of 3 different toppings (Nuts, Whipped Cream, or Fudge)
one of 2 types of cones (Sugar or Waffle)
How many different choices are there?
Theorem:
Multiplication Principle of Counting
If a task consists of a sequence of choices in which there are p selections for the first choice, q selections
for the second choice, and r selections for the third choice, and so on, then the task of making these
selections can be completed in p q r choices.
Permutation: An ORDERED arrangement of r objects chosen from m objects.
3 Types:
1.
The n objects are distinct (different), and repetition is allowed in the selection of r of the objects.
(distinct with repetition)
The number of ordered arrangements of r objects chosen from n distinct objects with repetition
allowed is:
Example:
nr
How many ways can you answer 6 true/false questions?
2.
The n objects are distinct, and repetition is NOT allowed in the selection of r of them,
where r  n
(distinct without repetition)
The number of permutations of r objects chosen from n distinct objects without repetition allowed
is given by the formula:
Example:
P  n, r  
n!
 n  r !
There are 6 six books setting in a pile. How many ways can you arrange 4 of them on a shelf?
3.
The n objects are not distinct, and we use all of them in the same arrangement
(not distinct)
The number of permutations of n objects which there are n1 of one kind, n2 of a second kind…
and nk of a k th kind is given by:
Example:
n!
n1 ! n2 ! ...nk !
How many ways can you arrange the word WINNER? (Arrangements do not have to make a word)
Combination: An arrangement WITHOUT REGARD TO ORDER of r objects selected from n distinct objects
without repetition, where r  n .
The number of combinations of r objects chosen from n distinct objects is given by the formula
C  n, r  
Example:
n
n!
 
 n  r !r !  r 
(this is a binomial coefficient)
4 out of 6 students are selected to go to the movie. How many ways can they be selected?
Examples – Use the correct formula to answer the following problems.
20. List all ordered arrangements of 6 objects:
22. List all combinations of 5 objects:
1,2,3,4,5, and 6, choosing 2 at a time without
a,b,c,d, and e, choosing 2 at a time. What
repetition. What is P  6, 2  ?
26.
A woman has 3 blouses and 5 skirts. How
Many different outfits can she wear?
is C  5, 2  ?
34.
How many different three-letter codes are there if
only the letters A,B,C,D, and E can be used and no
letter can be used more than once?
… if repetition is allowed?
50.
The student relations committee of a college
consists of 2 administrators, 3 faculty members,
and 5 students. Four administrators, 8 faculty
members, and 20 students are eligible to serve.
How many different committees are possible?
52.
How any different 11-letter words (real or
imaginary) can be formed from the letters in the
word MATHEMATICS?
54.
An urn contains 15 red balls and 10 white balls.
Five balls are selected. In how many ways can
the 5 balls be drawn from the total of 25 balls:
(a) if all 5 balls are red?
56.
A defensive football squad consists of 25 players.
Of these, 10 are linemen, 10 are linebackers, and
5 are safeties. How many different teams of 5
linemen, 3 linebackers, and 3 safeties can be
made?
(b) if 3 balls are red and 2 are white?
(c) if at least 4 are red?
Honors Pre-Calculus
12.3
Probability of Equally Likely Outcomes
Learning Targets: Students will be able to compute probability with equally likely outcomes.
Probability Model has:
1. Sample Space (a set S whose elements represent all possible outcomes of an experiment)
2. Probability assigned to each outcome
a. each probability is NON-NEGATIVE
b. the sum of the probabilities equal 1
When the same probability is assigned to each outcome of the sample space, the experiment is said to have equally likely
outcomes.
Example: Flip a coin - What is the probability that heads appears? What is the probability that tails appear?
Theorem: Probability of Equally Likely Outcomes
If an experiment has x equally likely outcomes, and if the number of ways that an event E can occur is y, the
P( E ) 
probability of E is:
n( E ) number of ways that E can occur y


n( S ) number of all logical possibilities x
Examples
12. Construct a probability MODEL for the experiment:
Tossing one fair coin three times
16.
Use the following spinners to construct a probability model for the experiment:
yellow
2
1
4
3
Spinner 1
green
Forward
red
Backward
Spinner 2
Spinner 3
Spin spinner 2, the spinner 1, and then spinner 3. What is the probability of getting Yellow, followed by a 2 or a 4,
followed by forward.
24.
A coin is weighted so that tails is twice as likely as heads to occur. What probability should we assign to heads and
to tails?
28.
Let the sample space be S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Supposed the outcomes are equally likely.
Compute the probability of the event F = {3, 5, 9, 10}
32.
An urn contains 5 white marbles, 10 green marbles, 8 yellow marbles, and 7 black marbles.
If one marble is selected, determine the probability that it is black.
Honors Pre-Calculus
12.3 continued
Compound & Binomial Probabilities
Learning Targets: Students will be able to compute compound and Binomial Probabilities.
Remember from Yesterday –
The probability of and event E occurring is: P( E ) 
n( E ) number of ways that E can occur

n( S ) number of all logical possibilities
Compound Probabilities – probability of MULTIPLE events
Theorem – Addition Rule
For any two events E and R, the probability that either E or F occurs is
P( E  F )  P( E )  P( F )  P( E  F )
Example: In a standard deck of cards, what is the probability that an Ace or Heart is selected?
Theorem - Addition Rule for Mutually Exclusive Events
If E and F are mutually exclusive events, then P  E  F   P  E   P  F 
Example:
42. Find P  A  B  IF P  A  .25, P  B   .45, P  A  B   .6
Recall from 12.1
The complement of set A, denoted A , is the set of all elements in the universal set that are not in A.
Complement of an Event
Let S denote the sample space of an experiment and let E denote an event. The complement of E, denoted
E , is the set of all outcomes in S that are not outcomes in E.
Example:
If you roll a die, let E = rolling an e, what is E ?
Theorem – Computing Probabilities of Complementary Events
 
 
P E  1  P  E  likewise P  E   1  P E
50.
In Chicago, there is a 4% probability that Memorial Day will have a low temperature in the 30s. What is
the probability that next Memorial Day will not have a low temperature in the 30s in Chicago?
52.
If a golf ball is selected at random from a container that has 9 white balls, 8 green balls, and 3 orange
balls. Find the probability that the golf ball is white or orange?
58.
The chart below indicates the results of a consumer survey of annual incomes in 100 households.
Income
$0-9,999
$10,000-19,999 $20000-29,999 $30,000-39,000 $40,000 or more
# of households
5
35
30
20
10
Use this chart to find the probability that a household has an annual income between$10,000 and $29,000 inclusive?
(Like 65) What is the probability that at least 2 people have the same birthday in a group of 7 people?
(assume 365 days in a year)
Theorem – Binomial Probability (NOT IN BOOK)
If a certain event is repeated with the same probability, denoted p, then the probability of r successes in n
n r
nr
trials is found by the formula:
  p 1  p 
r
36.
Determine the probability of having 2 girls and 2 boys in a 4-child family.
Ex
Find the probability of rolling exactly 3 fours in 10 rolls of one fair die.
70.
A fair coin is tossed 4 times
a) Find the probability that exactly 1 tail appears
b) Find the probability that no more than 1 tail appears
74.
In a shipment of 50 transformers, 10 are known to be defective. If 30 transformers are picked at random,
what is the probability that all 30 are non-defective?