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Chapter 14
From Randomness to Probability
Randomness & Probability Models:
Behavior is random if, while individual
outcomes are uncertain, for a large
number of repetitions, outcomes are
regularly distributed.
A situation in which we know what
outcomes could happen, but we don’t
know which particular outcomes will
happen is called a random
phenomenon.
Example: If I roll a die once, I
can’t predict with any certainty
what number it will land on, but
if I roll sixty times, I can expect
it to land on 1 ten times, 2 ten
times, 3 ten times, etc.
The probability of an outcome is the
proportion of times the outcome would
occur for a large number of repetitions
(in the long run).
Example: The probability of a die landing
on 4 is the proportion of times a die
lands on 4 for a large number of
repetitions.
The set of all possible outcomes of an event is
the sample space of the event.
Example: For the event “roll a die and observe
what number it lands on” the sample space
contains all possible numbers the die could
land on.
S = {1, 2, 3, 4, 5, 6 }
An event is an outcome (or a set of outcomes)
from a sample space.
Example 1: When flipping three coins, an
event may be getting the result THT. In this
case, the event is one outcome from the
sample space.
Example 2: When flipping three coins, an
event may be getting two tails. In this case, the
event is a set of outcomes (TTH,THT, HTT)
from the sample space.
An event is usually denoted by
a capital letter.
For example, call getting two
tails event A.
The probability of event A is
denoted P(A).
Probability Rules:
The probability of any event is between 0 and 1 .
A probability of 0 indicates the event will never
occur, and a probability of 1 indicates the
event will always occur.
If S is the sample space, then P(S) = 1, since
some outcome in the sample space is
guaranteed to occur.
The probability that event A does not occur is
one minus the probability that A does occur.
That A will not occur is called the complement
of A and is denoted P(AC).
P  AC   1 – P(A)
Example: When flipping two coins, the
probability of getting two heads is 0.25. The
probability of not getting two heads is
1 - .25 = .75.
The union of two or more events is the event
that at least one of those events occurs.
A
B
Addition Rule for the Union of Two Events:
P A or B  P A  B  P A  PB  P A and B
The intersection of two or more events is the
event that all of those events occur.
A
B
Multiplication Rule for the Intersection of Two Events:
P A and B  P A  B  P A PB | A
P  B | A 
P  B  A
P  A
Example: In our class of 36 students, we found that 25 students like
to listen to rap music, 22 students like to listen to alternative music,
and 16 students like to listen to both rap and alternative music.
Find the probabilities of the following events:
A: a student likes to listen to rap
B: a student likes to listen to alternative
A or B: a student likes to listen to rap or alternative
A and B: a student likes to listen to rap and alternative
Example: In our class of 36 students, we found that 25 students like
to listen to rap music, 22 students like to listen to alternative music,
and 16 students like to listen to both rap and alternative music.
Describe the following events:
Ac:
Bc:
Ac or Bc:
Ac and Bc:
If events A and B are disjoint or
mutually exclusive (they have no
outcomes in common), then the
probability that A or B occurs is the
probability that A occurs plus the
probability that B occurs.
P(A U B) = P(A) + P(B)
Example: Let event A be rolling a die and landing on an even
number, and event B be rolling a die and landing on an odd
number.
The outcomes for A are { 2, 4, 6 } and the
outcomes for B are { 1, 3, 5 }. These events
are disjoint because they have no outcomes in
common. So the probability of A or B (landing
on either an even or an odd number) equals
the probability of A plus the probability of B.
P  A or B  
3 3
 1
6 6
Events A and B are independent if knowing
that one occurs does not change the
probability that the other occurs.
Example: Roll a yellow die and a red die. Event
A is the yellow die landing on an even number,
and event B is the red die landing on an odd
number. These two events are independent
because the occurence of A does not change
the probability of B.
If events A and B are independent then
the probability of A and B equals the
probability of A times the probability of
B.
P  A and B  
P(A) X P(B)
Example: The probability that the yellow
die lands on an even number and the red
die lands on an odd number is:
3 3 1 1
1
P  A and B        .25
6 6
2 2
4
If events A and B are independent, then
their complements, Ac and Bc are also
independent and Ac is independent of
B.
The probability that event A occurs if we know for
certain that event B will occur is called
conditional probability.
The conditional probability of A given B is
denoted: P(A|B)
If events A and B are independent then knowing
that event B will occur does not change the
probability of A so for independent events:
P  A | B   P(A)
Example: When flipping a coin twice, what is the probability
of getting heads on the second flip if the first flip was a
head?
Event A: getting heads on first flip
Event B: getting heads on second flip
Events A and B are independent since the outcome of the
first flip does not change the probability of the second flip,
so…
P  A | B   P  A 
1
2
Are events A and B independent?
Suppose you draw one card from a standard
deck of cards.
A = you draw an ace
B = you draw a heart
Practice Problems
1) If A U B = S (sample space), P(A and Bc) = .2, and P(Ac) = .5,
then P(A) = ________.
2) Given: P(A) = .4, P(B) = .25, and P(A and B) = .1. What
is P(A|B)?
3) If P(A) = .24, P(B) = .62, and A and B are independent,
what is P(A or B)?
4) You play tennis with a friend, and from past experience,
you believe that the outcome of each match is
independent. For any given match you have a probability
of .6 of winning. What is the probability that you win at
least one of the next 4 matches?
5) Suppose that you have a torn tendon and are facing
surgery to repair it. The surgeon explains the risks to
you. Infection occurs in 3% of such operations, the repair
fails in 14%, and both infection and failure occur together
in 1%. What percent of these operations succeed and are
free from infection?
6) Suppose that for a group of consumers, the probability of
eating pretzels is .75 and the probability of drinking coke
is .65. Further suppose that the probability of eating
pretzels and drinking coke is .55. Determine if these two
events are independent.
p. 339: # 20 Stats Project
In a large Introductory Statistics lecture hall, the professor
reports that 55% of the students enrolled have never
taken a Calculus course, 32% have taken only one
semester of Calculus, and the rest have taken two or
more semesters of Calculus. The professor randomly
assigns students to groups of three to work on the
project for the course. What is the probability that the
first groupmate you meet has studied
a) two or more semesters of calculus?
b) some Calculus?
c) no more than one semester of Calculus?
p. 339: # 22 Another Project
You are assigned to be part of a group of three students
from the Intro Stats class described in Exercise 20. What
is the probability that of your other two groupmates,
a) neither has studied Calculus?
b)
both have studied at least one semester of Calculus?
c)
at least one has had more than one semester of
Calculus?
p. 340: # 26 Failing fathers?
A Pew Research poll in 2007 asked 2020 U.S. adults
whether fathers today were doing as good a job of
fathering as fathers of 20-30 years ago. Here’s how they
responded:
Response
Number
Better
424
Same
566
Worse
950
No Opinion
80
TOTAL
2020
If we select a respondent at random from this sample of
2020 adults, what is the probability that
a) the selected person responded “Worse?”
b) the person responded the “Same” or “Better?”