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Probability
Suppose an experiment (such as rolling a die, tossing a coin, or selecting a
candy from a bag of M&M’s) is repeated many times. The relative
frequency associated with an outcome (such as the die landing with a
particular value or heads rather than tails in the coin toss, or selection of a
yellow candy) will tend toward some number between 0 and 1 (inclusive)
as the number of repetitions increases. The long-term relative frequency
approaches the probability of the outcome.
A chance experiment is any activity or situation in which there is
uncertainty concerning which of two or more possible outcomes will result.
The probability of an outcome is interpreted as the long-run proportion of
the time that the outcome would occur, if the experiment were repeated
indefinitely. That is, probability is long-term relative frequency.
The Law of Large Numbers
As the number of repetitions of a probability experiment increases, the
proportion with which a certain outcome is observed gets closer to the
probability of the outcome.
Examples:
P(Head) = 1/2 in the experiment of tossing a coin
P (2) = 1/6 in the experiment of rolling a die
Sample Space is the set of all possible outcomes.
Examples:
S= {H, T} is the sample space while tossing a coin
S= {1,2,3,4,5,6} is the sample space while rolling a die
Consider the probability experiment of having two children.
(a) Identify the outcomes of the probability experiment.
(b) Determine the sample space.
(c) Define the event E = “have one boy”.
(a)
(b)
(c)
e1 = boy, boy, e2 = boy, girl, e3 = girl, boy, e4 = girl, girl
{(boy, boy), (boy, girl), (girl, boy), (girl, girl)}
{(boy, girl), (girl, boy)}
Rules for Probability
1. The probability of each outcome must be a number between 0 and 1,
inclusive.
2. The probabilities of all of the outcomes in a given sample space must
add up to one.
A probability model lists the possible outcomes of a probability
experiment and each outcome’s probability. A probability model must
satisfy rules 1 and 2 of the rules of probabilities.
Calculating the probability for an event
An event is a specified set of possible outcomes in the sample space.
A – An even number shows
A= {2,4,6}
The probability of an event equals the sum of the probabilities associated
with the outcomes that constitute that event.
P(A)=1/6+1/6+1/6=1/2
If an event is impossible, the probability of the event is 0.
If an event is a certainty, the probability of the event is 1.
An unusual event is an event that has a low probability of occurring.
Probability Model example:
In a bag of peanut M&M milk chocolate candies, the colors of the candies
can be brown, yellow, red, blue, orange, or green. Suppose that a candy is
randomly selected from a bag. The table shows each color and the
probability of drawing that color. Verify this is a probability model.
Color Probability
Brown
0.12
Yellow
0.15
Red
0.12
Blue
0.23
Orange
0.23
Green
0.15
1. All probabilities are between 0 and 1, inclusive.
2. Because 0.12 + 0.15 + 0.12 + 0.23 + 0.23 + 0.15 = 1, rule 2 (the sum of
all probabilities must equal 1) is satisfied.
Approximating Probabilities Using the Empirical Approach
The probability of an event E is approximately the number of times event E
is observed divided by the number of repetitions of the experiment.
P(E) ≈ relative frequency of E

frequency of E
number of trials of experiment
The classical method of computing probabilities requires equally likely
outcomes.
An experiment is said to have equally likely outcomes when each
outcome has the same probability of occurring.
Computing Probability Using the Classical Method
So, if S is the sample space of this experiment,
N E 
P E  
N S 
where N(E) is the number of outcomes in E, and N(S) is the number of
outcomes in the sample space.
Computing Probability using Classical approach :
Suppose a “fun size” bag of M&Ms contains 9 brown candies, 6 yellow
candies, 7 red candies, 4 orange candies, 2 blue candies, and 2 green
candies. Suppose that a candy is randomly selected.
(a) What is the probability that it is yellow?
(b) What is the probability that it is blue?
(c) Comment on the likelihood of the candy being yellow versus blue.
(a)
There are a total of 9 + 6 + 7 + 4 + 2 + 2 = 30 candies
N(S) = 30.
N(yellow)
N(S)
6

 0.2
30
P(yellow) 
(b)P(blue) = 2/30 = 0.067.
(c) Since P(yellow) = 6/30 and P(blue) = 2/30, selecting a yellow is
three times as likely as selecting a blue.
Mutually Exclusive Events
Two Events A and B are mutually exclusive if when an experiment is
conducted a single time, the occurrence of one event excludes the possibility
of the occurrence of the other event.
A – 1 shows on the die
B – 6 shows on the die
P(A or B)= 1/6 + 1/6=2/6=1/3
P(A or B)= P(A)+P(B) for mutually exclusive events.
In general, the addition rule is
P (A or B) = P(A) + P(B) – P (A and B)
If events A and B are mutually exclusive, P (A and B) = 0.
Suppose that a pair of dice are thrown. Let E = “the first die is a two” and
let F = “the sum of the dice is less than or equal to 5”. Find P (E or F)
using the General Addition Rule.
N(E)
N(S)
6

36
1

6
P(E) 
N(F)
N(S)
10

36
5

18
P(F) 
P(E and F
N (E and F)

N (S)
3

36
1

12
P(E or F)  P(E)  P(F)  P(E and F)
6 10 3



36 36 36
13

36
Complement of an Event
Let S denote the sample space of a probability experiment and let E denote
an event. The complement of E, denoted EC, is all outcomes in the sample
space S that are not outcomes in the event E.
Complement Rule
If E represents any event and EC represents the complement of E, then
P(EC) = 1 – P(E)
Example: According to the American Veterinary Medical Association,
31.6% of American households own a dog. What is the probability that a
randomly selected household does not own a dog?
P(do not own a dog) = 1 – P(own a dog)
= 1 – 0.316
= 0.684
Independent Events
Two events E and F are said to be independent if the probability of E
occurring is not affected by event F having occurred or vice versa.
If E and F are not independent, they are said to be dependent events
Examples:
(a)Suppose you draw a card from a standard 52-card deck of cards and then roll a
die. The events “draw a heart” and “roll an even number” are independent
because the results of choosing a card do not impact the results of the die toss.
(b)
Suppose two 40-year old women who live in the United States are
randomly selected. The events “woman 1 survives the year” and “woman 2
survives the year” are independent.
(c)Suppose two 40-year old women live in the same apartment complex. The
events “woman 1 survives the year” and “woman 2 survives the year” are
dependent
Multiplication Rule for Independent events:
If two events E and F are independent, then P (E and F) =P(E) * P(F).
A manufacturer of exercise equipment knows that 10% of their products are
defective. They also know that only 30% of their customers will actually use the
equipment in the first year after it is purchased. If there is a one-year warranty on the
equipment, what proportion of the customers will actually make a valid warranty
claim?
We assume that the defectiveness of the equipment is independent of the use of the
equipment. So,
P defective and used   P defective  P used 
 (0.10)(0.30)
 0.03
Multiplication Rule for n Independent Events:
If E1, E2, E3, … and En are independent events, then
P E1 and E2 and E3 and ... and En 
 P E1  P E2    P En 
The probability that a randomly selected female aged 60 years old will survive the
year is 99.186% according to the National Vital Statistics Report, Vol. 47, No. 28.
What is the probability that four randomly selected 60 year old females will survive
the year?
P(all 4 survive)
= P(1st survives and 2nd survives and 3rd survives and 4th survives)
= P(1st survives) . P(2nd survives) . P(3rd survives) . P(4th survives)
= (0.99186) (0.99186) (0.99186) (0.99186)
= 0.9678
The probability that a randomly selected female aged 60 years old will survive the
year is 99.186% according to the National Vital Statistics Report, Vol. 47, No. 28.
What is the probability that at least one of 500 randomly selected 60 year old
females will die during the course of the year?
Hint: Use complement Rule
P (at least one dies) = 1 – P (none die)
= 1 – P (all survive)
= 1 – (0.99186)500
= 0.9832
Conditional Probability:
Let E and F be two events with P(F)>0.
The conditional probability of the event E given that the event F has
occurred, denoted by P(E|F), is
and F )
P(E|F)= P( EP(F)
Example: A survey was conducted by the Gallup Organization conducted May 8 –
11, 2008 in which 1,017 adult Americans were asked, “Which of the following
statements comes closest to your belief about God – you believe in God, you don’t
believe in God, but you do believe in a universal spirit or higher power, or you
don’t believe in either?”
Believe in
God
Believe in
universal spirit
Don’t believe
in either
East
204
36
15
Midwest
212
29
13
South
219
26
9
West
152
76
26
(a)What is the probability that a randomly selected adult American who lives in
the East believes in God?
(b) What is the probability that a randomly selected adult American who believes
in God lives in the East?
(a)
P believes in God lives in the east  

N believe in God and live in the east 
N live in the east 
204
 0.8
204  36  15
(b)
P lives in the east believes in God 


N believe in God and live in the east 
N believes in God 
204
 0.26
204  212  219  152
General Multiplication Rule:
The probability that two events E and F both occur is
P E and F   P E  P F E 
In words, the probability of E and F is the probability of event E occurring
times the probability of event F occurring, given the occurrence of event
E.
In 2005, 19.1% of all murder victims were between the ages of 20 and 24 years old.
Also in 2005, 86.9% of murder victims were male given that the victim was 20 – 24
years old. What is the probability that a randomly selected murder victim in 2005
was a 20 – 24-year-old male?
P male and 20  24   P 20  24  P male 20  24 
 0.869  0.191  0.165979  0.166