Transcript STATISTICS I

STATISTICS I
COURSE INSTRUCTOR:
TEHSEEN IMRAAN
CHAPTER 5
PROBABILITY
TYPES OF STATISTICS
• DESCRIPTIVE STATISTICS
• INFERENTIAL STATISTICS
WHAT IS PROBABILITY?
• A value between zero and one, inclusive, describing the
relative possibility (chance or likelihood) an event will
occur.
• Probability is expressed either as a percent or as a decimal.
The likelihood that any particular event will happen may
assume values between 0 and 1.0. A value close to 0
indicates the event is unlikely to occur, whereas a value
close to 1.0 indicates that the event is quite likely to occur.
• To illustrate, a value of 0.60 might express your degree of
belief that tuition will be increased at your college, and
0.50 the likelihood that your first marriage will end in
divorce.
EXPERIMENT, OUTCOME & EVENT
• Experiment: A process that leads to the
occurrence of one and only one of several
possible observations.
• Outcome: A particular result of an
experiment.
• Event: A collection of one or more outcomes
of an experiment.
APPROACHES TO PROBABILITY
CLASSICAL PROBABILITY
• A probability based on the assumption that
outcomes of an experiment are equally likely.
• To find the probability of a particular outcome
we divide the number of favorable outcomes
by the total number of possible outcomes as
shown in text formula [5-1].
CLASSICAL PROBABILITY
CLASSICAL PROBABILITY
For example, you take a multiple-choice
examination and have no idea which one of
the choices is correct. In desperation you
decide to guess the answer to each question.
The four choices for each question are the
outcomes. They are equally likely, but only
one is correct. Thus the probability that you
guess a particular answer correctly is 0.25
found by 1/ 4.
MUTUALLY EXCLUSIVE
The occurrence of one event means that none
of the other events can occur at the same
time. For instance, an employee selected at
random is either a male or female but cannot
be both. A computer chip cannot be defective
and not defective at the same time.
COLLECTIVELY EXHAUSTIVE
• If an experiment has a set of events that includes
every possible outcome, then the set of events is
called collectively exhaustive.
• At least one of the events must occur when an
experiment is conducted.
• For example: In a die-tossing experiment every
outcome will be either an even number or an odd
number. Thus the set is collectively exhaustive. If
the set of events is collectively exhaustive and the
events are mutually exclusive, the sum of the
probabilities equals 1.
EMPIRICAL CONCEPT
• Another way to define probability is based on relative
frequencies. The probability of an event happening is
determined by observing what fraction of the time similar
events happened in the past.
• To find a probability using the relative frequency approach
we divide the number of times the event has occurred in
the past by the total number of observations. Suppose the
Civil Aeronautics Board maintained records on the number
of times flights arrived late at the Newark International
Airport. If 54 flights in a sample of 500 were late, then,
according to the relative frequency formula, the probability
a particular flight is late is 0.108.
• Based on passed experience, the probability is 0.108 that a
flight will be late.
EMPIRICAL CONCEPT
SUBJECTIVE PROBABILITY
• The likelihood (probability) of a particular
event happening that is assigned by an
individual based on whatever information is
available.
• Subjective probability is based on judgment,
intuition, or "hunches." The likelihood that the
horse, Sir Homer, will win the race at Ferry
Downs today is based on the subjective view
of the racetrack odds maker.
Some rules for assigning probabilities
In the study of probability it is often necessary
to combine the probabilities of events. This is
accomplished through both rules of addition
and rules of multiplication. There are two
rules for addition,
• the special rule of addition and
• the general rule of addition.
Special Rule for Addition
• To apply the special rule of addition, the events
must be mutually exclusive. The special rule of
addition states that the probability of the event A
or the event B occurring is equal to the
probability of event A plus the probability of
event B. The rule is expressed by using text
formula [5-2]
• To apply the special rule of addition the events
must be mutually exclusive. This means that
when one event occurs none of the other events
can occur at the same time.
Special Rule for Addition
Venn Diagram
• Venn diagrams, developed by English logician
J. Venn, are useful for portraying events and
their relationship to one another. They are
constructed by enclosing a space, usually in a
form of a rectangle, which represents the
possible events. Two mutually exclusive
events such as A and B can then be portrayed
as in the following diagram by enclosing
regions that do not overlap (that is, that have
no common area).
Venn Diagram
The Complement Rule
• The complement rule is used to determine the
probability of an event occurring by subtracting
the probability of the event not occurring from
one
• Complement rule: A way to determine the
probability of an event occurring by subtracting
the probability of an event not occurring from 1.
• In some situations it is more efficient to
determine the probability of an event happening
by determining the probability of it not
happening and subtracting from 1.
The Complement Rule
General Rule of Addition
• When we want to find the probability that two
events will both happen, we use the concept
known as joint probability.
• Joint probability: A probability that measures the
likelihood two or more events will happen
concurrently.
• What if the events are not mutually exclusive? In
that case the general rule of addition is used. The
probability is computed using the text formula [54].
General Rule of Addition
General Rule of Addition
Where:
• P(A) is the probability of the event A.
• P(B) is the probability of the event B.
• P(A and B) is the probability that both events
A and B occur.
General Rule of Addition
• For example, a study showed 15 percent of
the work force to be unemployed, 20 percent
of the work force to be minorities, and 5
percent to be both unemployed and
minorities. What percent of the work force are
either minorities or unemployed?
Solution
P (unemployed or minority) = P (unemployed) +
P (minority) - P (unemployed and minority)
= 0.15 + 0.20 - 0.05
= 0.30
Venn diagram
Using addition rule
Rules of Multiplication
• There were two rules of addition, the general
rule and the special rule. We used the general
rule when the events were not mutually
exclusive and the special rule when the events
were mutually exclusive. We have an
analogous situation with the rules of
multiplication.
Special Rule of Multiplication
• We use the general rule of multiplication when the two
events are not independent and the special rule of
multiplication when the events A and B are independent.
Two events are independent if the occurrence of one does
not affect the probability of the other.
• A way to look at independence is to assume two events A
and B occur at different times. For example, flipping a coin
and getting tails is not affected by rolling a die and getting a
two.
• The special rule of multiplication is used to combine events
where the probability of the second event does not depend
on the outcome of the first event.
Conditional Probability
• The probability of a particular event occurring
may be altered by another event that has already
occurred.
• Probability measures uncertainty, but the degree
of uncertainty changes as new information
becomes available. Symbolically, it is written P
(B\A). The vertical line (\) does not mean divide;
it is read "given that" as in the probability of B
"given that" A already occurred.
Conditional Probability
• Rule of Independence: P(A)= P(A\B)
• The rule of independence restates our
definition. It says that the probability of event
A is equal to the conditional probability of A
given B. In other words event A is unaffected
by any prior occurrence of event B.
Special Rule of Multiplication
• P(A and B) = P(A) * P(B)
• As an example, a nuclear power plant has two independent safety
systems. The probability the first will not operate properly in an
emergency P (A) is 0.01, and the probability the second will not
operate P (B) in an emergency is 0.02. What is the probability that
in an emergency both of the safety systems will not operate? The
probability both will not operate is:
• The probability 0.0002 is called a joint probability, which is the
simultaneous occurrence of two events. It measures the likelihood
that two (or more) events will happen together (jointly).
SPECIAL RULE OF MULTIPLICATION IN
CASE OF THREE EVENTS
• The probability for three independent events,
A, B, and C, the special rule of multiplication
used to determine the probability of all three
events will occur is:
P(A and B and C)= P(A).P(B).P(C)
GENERAL RULE OF MULTIPLICATION
• The general rule of multiplication is used to
combine events that are not independent that is,
they are dependent on each other. For two
events, the probability of the second event is
affected by the outcome of the first event. Under
these conditions, the probability of both A and B
occurring is given in formula [5-7].
• P(A and B)= P(A) . P(B\A),
where P (B\A ) is the probability of B occurring
given that A has already occurred. Note that P
(B\A) is a conditional probability.
GENERAL RULE OF MULTIPLICATION
• For example, among a group of twelve
prisoners, four had been convicted of murder.
If two of the twelve are selected for a special
rehabilitation program, what is the probability
that both of those selected are convicted
murderers?
Solution
• Let A1 be the first selection (a convicted
murderer) and A2 the second selection (also a
convicted murderer). Then P(A1) = 4/12. After
the first selection, there are 11 prisoners, 3 of
whom are convicted of murder, hence
P(A2\A1) = 3/11. The probability of both A1 and
A2 happening is:
Solution
Contingency Table
• A table used to classify sample observations according
to two or more identifiable characteristics
• A contingency table is a cross tabulation that
simultaneously summarizes two variables of interest
and their relationship.
• A survey of 200 school children classified each as to
gender and the number of times Pepsi-Cola was
purchased each month at school. Each respondent is
classified according to two criteria-the number of times
Pepsi was purchased and gender.
Contingency Table
Gender
Bought
Boys
Pepsi
Girls
Total
0
5
10
15
1
15
25
40
2 or
more
80
65
145
Total
100
100
200
Using the multiplication rule
Tree diagram
• The tree diagram is a graph that is helpful in
organizing the calculations that involve several
stages. Each segment in the tree is one stage
of the problem. The branches are weighted
probabilities.
Principles of Counting
• If the number of possible outcomes in an
experiment is small, it is relatively easy to count
the possible outcomes. However, sometimes the
number of possible outcomes is large, and listing
all the possibilities would be time consuming,
tedious, and error prone. Three formulas are very
useful for determining the number of possible
outcomes in an experiment. They are: the
multiplication formula, the permutation
formula, and the combination formula.
Multiplication Rule
• Multiplication formula: If there are m ways of
doing one thing, and n ways of doing another
thing, there are m x n ways of doing both.
• In terms of formula, it is
total no. of arrangements= (m).(n)
Permutation Rule
• The permutation is an arrangement of objects
or things wherein order is important. That is,
each time the objects or things are placed in a
different order, a new permutation results.
• Any arrangements of r objects selected from a
single group of n possible objects.
Permutation Formula
Permutation Formula,
Permutation Formula
• Where:
• P is the number of permutations, or ways the
objects can be arranged.
n is the total number of objects.
r is the number of objects selected.
• Note: the n! is a notation called "n factorial."
• For example, 4! means 4 times 3 times 2 times
1 = 4 × 3 × 2 × 1 = 24
Combination Rule
• One particular arrangement of the objects
without regard to order is called a
combination.
• The number of ways to choose r objects from
a group of n possible objects without regard
to order.
Combination Formula
Combination Formula,
Combination Formula
• Where:
• C is the number of different combinations.
n is the total number of objects.
r is the number of objects to be used at one
time.