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Introduction to Probability
Event: something that can occur (picking a marble out of
a jar, playing a game of basketball, choosing people for a
committee, etc.) that yield specific outcomes.
Independent Events: Two events whose outcomes
do not affect each other.
Basic Counting Principle: Suppose that one event can be
chosen in ‘p’ different ways, and another independent event
can be chosen in ‘q’ different ways. Then the two events can
be chosen successively in ‘p * q’ ways.
Basic Counting Principle
Example of the Basic Counting Principle: At a particular fast
food restaurant, there are 8 different hamburgers to choose from,
5 different soda flavors, and the fries can be salted, unsalted, or
cheesy. How many different ways are there to pick a burger,
soda, and fries?
Basic Counting Principle
Example of the Basic Counting Principle: A quarter, a nickel, a dime,
and a penny are tossed simultaneously. How many different ways
can the coins land?
Permutations
Permutation: Arrangement of objects in a certain order.
The Number of Permutations of n objects, taken r at a time,
is defined as follows…
nPr
=
n!
(n – r)!
Remember, n! = n*(n-1)*(n-2)*…* 2*1
Permutations
Example of a permutation: How many ways can a novel, an atlas,
a dictionary, and an encyclopedia be placed on a bookshelf?
Permutations
Example of a permutation: How many ways can you create
A three letter arrangement of the letters in the word ‘keyboard’?
Permutations
Example of a permutation: How many ways a license plate
be made in Pennsylvania, assuming that the plate must have
3 letters and then 4 numbers, without repeating any number
or letter?
Permutations
Example of a permutation: How many ways a license plate
be made in Pennsylvania, assuming that the plate must have
3 letters and then 4 numbers, if repetition is allowed?
Permutation Problem
Six different Spanish books and 4 different French
books are to be placed on a shelf with the French
books together and at the right of the Spanish books.
How many ways can this be done?
Permutations with
Repetitions and Circular Permutations
Permutation with Repetition: Arrangement of objects in a
certain order, where two or more of the objects are alike.
The Number of Permutations of n objects, taken n at a time,
of which p are alike and q are alike, is defined as follows…
n!
p!q!
Permutations with
Repetitions and Circular Permutations
Example: How many ways can you arrange the letters in the
word “Seattle”?
Permutations with
Repetitions and Circular Permutations
Example: How many ways can you arrange the WORDS in the
phrase “Take me out to the ballgame, take me out with the crowd”?
Permutations with
Repetitions and Circular Permutations
Circular Permutations: Arrangement of objects in a certain
order around a circle (where there is no beginning or end).
The Number of Permutations of n objects, taken n at a time,
given that the n objects are arranged in a circle, is defined as…
n!
n
= (n-1)!
Permutations with
Repetitions and Circular Permutations
Example: How many ways can 5 people be arranged about
a round table?
Permutations with
Repetitions and Circular Permutations
Example: How many ways can you arrange 5 people around
a table, relative to the door of the room?
Permutations with
Repetitions and Circular Permutations
When a permutation is reflexive, there are
actually half as many permutations available as before.
Example: How many ways can you arrange 6 beads on a
necklace with no clasp?
Permutations with
Repetitions and Circular Permutations
How many ways can the letters in the word
‘OKLAHOMA’ be arranged?
Combinations
Combination: The selection of objects without regard to the
order in which the objects are picked.
The Number of Combinations of n objects, taken r at a time,
(which can be written as C(n,r) ) is defined as follows…
nCr
= n!
(n – r)!r!
Combinations
Example: How many ways can 4 students be chosen for
a committee, out of 15 possible students?
Combinations
Example: How many ways can a 5 card hand be dealt?
(assuming a standard deck of playing cards, which has 52 cards)
Combinations
Example: A bag of marbles contains 5 red, 6 white, and
4 green marbles. How many ways can 3 red and 2 white
marbles be chosen?
Combinations
Example: Suppose that from a group of 7 men and 9 women,
a committee of 4 is to be formed. How many committees can
be formed if the committee contains 2 men and 2 women?
Combinations
In how many ways can 5 basketball players be
chosen from 9 seniors and 5 juniors if all are
equally eligible?
If the team must have 2 seniors and 3 juniors?
Combinations
Find the number of triangles determined by 9
points on a circle
Combinations
Out of a group of 10 sophomores, 3 juniors, 4
seniors, and 5 freshmen, a committee of 4
needs to be selected. How many ways are
possible if…
a) There are no restrictions to who can be on the
committee
b) There must be 2 juniors and 2 seniors
c) The committee must be made entirely of
freshmen
Probability Problem
There are 16 members in the Debate Club. How
many ways can a President, Vice President,
Secretary, and Treasurer be chosen?
Probability
Probability: The likelihood that a given event will occur.
If an event can succeed in ‘s’ ways and fail in ‘f’ ways, then
the probability of success and the probability of failure are
determined as follows:
P(s) =
s
s+f
P(f) =
f
s+f
Since an event can either succeed or fail, and since the
highest probability is ‘1’, then for any event,
P(s) + P(f) = 1
Probability
Odds: The odds of a successful outcome of an event is the
ratio of the probability of its success to the probability
of its failure.
Odds = P(s)
P(f)
Probability
Example: Determine the probability of picking a diamond
out of a standard deck of cards.
Example: Determine the odds of picking a diamond
out of a standard deck of cards.
Probability
Example: Determine the probability of rolling a 5 on
a die.
Example: What are the odds of rolling a 5 on
a die?
Probability
Example: On the volleyball team, there are 5 freshmen,
6 sophomores, 9 juniors, and 4 seniors. If one is selected
at random, what is the probability that the student is a
freshman?
Example: What are the odds that the student is a
freshman?
Probability
Example: Suppose two letters are picked at random from
the word ‘computer’. What is the probability that both of
the letters are consonants?
Example: What are the odds that both letters are
consonants?
Probability
Example: Two dice are thrown. Find the probability that
both dice show the same number.
Probability
One card is drawn at random from an ordinary deck of
cards. Find the probability of each event.
a.
a red face card
b.
6 of clubs
c.
a king
d.
not a king
Probability
A number wheel is divided in 12 equal sectors,
numbered 1 – 12. Find the probability of each event
a.
It is a 5
b.
It is a 2-digit number
c.
It is a factor of 12
d.
It is an odd number
e.
It is a negative number
f.
It is a 3 or a 6 or a 9
Probability
1.
A letter is selected at random from those in the
word
RECTANGLE. Find the probability of each event.
a. It is a vowel
b.
It is a consonant
c.
It is between D and M in the alphabet
d.
It is either C or G
Probability of Independent
and Dependent Events
Probability of Two Independent Events: If two events,
A and B, are both independent, then the probability of
both events occurring is…
P(A and B) = P(A)*P(B)
Probability of Independent
and Dependent Events
Probability of Two Dependent Events: If two events,
A and B, are dependent, then the probability of
both events occurring is…
P(A and B) = P(A) * P(B following A)
Probability of Independent
and Dependent Events
Example of Dependent Events: What is the probability of
drawing 6 hearts from a deck of cards without replacement?
Probability of Independent
and Dependent Events
Example of Independent Events: What is the probability of
rolling a 4 on a die 3 times in a row?
Probability of Independent
and Dependent Events
Suppose the odds of the Sixers beating the Kings in
Basketball was 5 : 2. What is the probability of
the Sixers beating the Kings 4 times in a row?
Probability of Independent
and Dependent Events
A particular bag of marbles contains 4 red, 6 green,
2 blue, and 5 white marbles. What is the probability
of picking a red, white, and blue marble, in that order?
What would the probability be with replacement?
Probability of Mutually
Exclusive Events and Inclusive Events
Mutually Exclusive Events: If two events, A and B, are
mutually exclusive, then that means that if A occurs, than
B cannot, and vice versa.
Probability of Mutually Exclusive Events: If two events,
A and B, are mutually exclusive, then the probability that
either A OR B occurs is…
P(A or B) = P(A) + P(B)
Probability of Mutually
Exclusive Events and Inclusive Events
Inclusive Events: If two events, A and B, are inclusive,
then that means that if A occurs, B could also occur,
and vice versa.
Probability of Inclusive Events: If two events, A and B, are
inclusive, then the probability that either A or B occurs is…
P(A or B) = P(A) + P(B) – P(A and B)
Probability of Mutually
Exclusive Events and Inclusive Events
Example: A particular bag of marbles contains 4 red,
6 green, 2 blue, and 5 white marbles. If 3 marbles are
picked, what is the probability of picking all reds or
all greens?
Probability of Mutually
Exclusive Events and Inclusive Events
Example: Slips of paper numbered 1 to 15 are placed
in a box. A slip of paper is drawn at random. What is
the probability that the number picked is either a
multiple of 5 or an odd number?
Probability of Mutually
Exclusive Events and Inclusive Events
Example: 4 coins are tossed. What is the probability
of obtaining 2 heads or 1 tail?
Probability of Mutually
Exclusive Events and Inclusive Events
Example: Two cards are picked out of a standard deck.
What is the probability of picking of both cards being
either face cards or clubs?
Conditional Probability
Conditional Probability: The conditional probability of
event A, given event B, is found as follows…
P(A/B) =
P(A and B)
P(B)
A sample space is the set of all outcomes for an event.
A reduced sample space is one that contains only those
outcomes that satisfy a given condition.
Conditional Probability
Example: A card is drawn from a standard deck. What
is the probability that the card is a face card, if it is
known to be a diamond?
Conditional Probability
Conditional Probability: Sally tosses 3 coins. What is
the probability that she has tossed exactly 2 heads, given
that she has tossed at least one head?
Conditional Probability
Example: A die is thrown four times. What is the
Probability that a ‘2’ shows on the die the fourth time,
Given that a 2 showed up the first three times?
Conditional Probability
Example: A team of 5 people is chosen from these
basketball players: John, Sam, Jim, Steve, Ray, Rick,
Charlie, Bill, and Doug. If Sam is on the team, what is
the probability that Bill is also on the team?
Probability Review Questions
At a given university, the student ID number is an 8
digit number. The first two digits indicate the year of
graduation, and the rest are all non-zero digits. For
the class of 2006, how many student ID numbers are
possible?
Probability Review Questions
Four cards are picked at random from a standard deck.
What is the probability that all four cards are face cards?
Thirty tickets are numbered 1 to 30 consecutively.
What is the probability that 4 prime numbers are
chosen if replacement is allowed?
Probability Review Questions
How many arrangements of letters can be made from the
word “commencement”?
The S.G.A. has 5 freshmen members, 6 sophomores,
6 juniors, and 9 seniors. How many ways can a special
Committee of five be selected if it must have…
a) 3 seniors and 1 junior
b) 2 seniors and 1 each from the other grade levels
Probability Review Questions
The Debate Team has a competition coming up and they
need to pick 3 out of all their members to represent them
in the contest. They have 4 freshmen, 2 sophomores,
3 juniors, and 5 seniors on their team. If the 3 members
are picked at random for this competition, what is the
probability that the team will have at least 2 juniors?
Probability Review Questions
Suppose you pick 3 cards at random out of a standard
deck. What is the probability that all 3 are odd
numbered cards?
Probability Review Questions
A committee of 3 is to be picked from the group of
Donna, Patty, Pam, Jim, Barb, and Luke. What is the
probability that Pam will be chosen for the committee
if Barb is not on the committee?
Probability: Monty Hall Problem
Let's Make a Deal!
Imagine that the set of Monty Hall's game show Let's Make a Deal has three closed
doors. Behind one of these doors is a car; behind the other two are goats. The
contestant does not know where the car is, but Monty Hall does. The contestant
picks a door and Monty opens one of the remaining doors, one he knows doesn't
hide the car. If the contestant has already chosen the correct door, Monty is
equally likely to open either of the two remaining doors. After Monty has shown
a goat behind the door that he opens, the contestant is always given the option to
switch doors.
What is the probability of winning the car if she stays with her first choice?
What if she decides to switch?
Probability: Monty Hall Problem
Let's Make a Deal!
We will assume that there is a winning door and that the two remaining doors,
A and B, both have goats behind them. There are three options:
1.
The contestant first chooses the door with the car behind it. She is then shown
either door A or door B, which reveals a goat. If she changes her choice of
doors, she loses. If she stays with her original choice, she wins.
2.
The contestant first chooses door A. She is then shown door B, which has a
goat behind it. If she switches to the remaining door, she wins the car.
Otherwise, she loses.
3.
The contestant first chooses door B. She is then is shown door A, which has a
goat behind it. If she switches to the remaining door, she wins the car.
Otherwise, she loses.
Probability: Monty Hall Problem
Let's Make a Deal!
Each of the above three options has a 1/3 probability of occurring, because the
contestant is equally likely to begin by choosing any one of the three doors. In two
of the above options, the contestant wins the car if she switches doors; in only one
of the options does she win if she does not switch doors. When she switches, she
wins the car twice (the number of favorable outcomes) out of three possible
options (the sample space).
Thus the probability of winning the car is 2/3 if she switches doors, which means
that she should always switch doors.