1. What do you mean by probably ? meaning not sure ! Or are
un-certain e.g. probably it will rain today or probably Pakistan
will win the match. In such and many other situations, there is an
element of un-certainty in our statements. In Statistics, we have a
technique or tool which is used to measure the amount of
un-certainty. i.e. PROBABILITY. 2. The range of this numerical
measure is from zero to one. i.e. if Ei is any event, then 0 P(Ei)
1 for each i The word Probability has two basic meanings: 1: It is
the quantitative measure of un-certainty 2: Its the measure of
degree of belief in a particular statement. Example of a melon
picked from a carton. 3. The first meaning is related to the
problems of objective approach. The second is related to the
problems of subjective approach. 4. There is a situation, where
probability is an inherent part e.g. we have: Population Sample
Here we have full Here we have Information and have partial
information certain results. and have un-certain results. 5. As we
know, mostly the problems are solved using the sample data. A
statistical technique which deals with the sample Results is
statistical inference. Un-certainty (due to sample data) is also an
inherent part of statistical inference. 6. Statistics and
Probability theory constitutes a branch of mathematics for dealing
with uncertainty Probability theory provides a basis for the
science of statistical inference from data 7. a random experiment
is an action or process that leads to one of several possible
outcomes. For example: 6. 8 Experiment Outcomes Flip a coin Heads,
Tails Exam Marks Numbers: 0, 1, 2, ..., 100 Roll a die 1,2,3,4,5,6
Course Grades F, D, C, B, A, A+ 8. List the outcomes of a random
experiment List: Called the Sample Space Outcomes: Called the
Simple Events This list must be exhaustive, i.e. ALL possible
outcomes included. Die roll {1,2,3,4,5} Die roll {1,2,3,4,5,6} 6.9
9. The list must be mutually exclusive, i.e. no two outcomes can
occur at the same time: Die roll {odd number or even number} Die
roll{ number less than 4 or even number} 10. A list of exhaustive
[dont leave anything out] and mutually exclusive outcomes
[impossible for 2 different events to occur in the same experiment]
is called a sample space and is denoted by S. 11. A usual six-sided
die has a sample space S={1,2,3,4,5,6} If two dice are rolled ( or,
equivalently, if one die is rolled twice) The sample space is shown
in Figure 1.2. 12. An individual outcome of a sample space is
called a simple event [cannot break it down into several other
events], An event is a collection or set of one or more simple
events in a sample space. 6.14 13. Roll of a die: S = {1, 2, 3, 4,
5, 6} Simple event: the number 3 will be rolled Compound Event: an
even number (one of 2, 4, or 6) will be rolled 14. The probability
of an event is the sum of the probabilities of the simple events
that constitute the event. E.g. (assuming a fair die) S = {1, 2, 3,
4, 5, 6} and P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6 Then:
P(EVEN) = P(2) + P(4) + P(6) = 1/6 + 1/6 +1/6 = 3/6 = 1/2 6.16 15.
Types of an Event Impossible Event Possible Event Simple Event
Compound Event Certain Event Mutually Exclusive Equally Likely
Exhaustive Dependent Independent 16. 17. A desired subset of a
sample space containing at least one possible outcome. Example When
a die is rolled once, Sample Space is S={ 1,2,3,4,5,6} Event E is
defined as an odd number appears on upper side of a die E={1,3,5}
18. A desired subset of a sample space containing only one possible
outcome. Example Event D is defined as head will turn up on a coin
D={ H} 19. A desired subset of a sample space containing at least
two possible outcome. Example Event E is defined as an odd number
appears on upper side of a die E={1,3,5} 20. A desired sub set of
sample space containing all possible outcomes. This event is also
called certain event. Example Any number from 1 to 6 appears on
upper side of a die. S={1,2,3,4,5,6} 21. Mutually Exclusive Events
Two events A and B defined in a sample space S and have nothing
common between them are called mutually exclusive events. Example
The events A (number 4 appear on a die) and B( an odd number appear
on a die) are two mutually exclusive events. 22. Equally Likely
Events Two events A and B defined in simple sample space S and
their chances of occurrences are equal are called equally likely
events. Example The events A (head will turn up on a coin) and B
(tail will turn up on a coin). 23. Two events A and B defined in
simple sample space S (i) Nothing common between them. (ii) Their
union is same as the sample space. Example The events A(head will
turn up on a coin) and B ( tail will turn up on a coin) are two
exhaustive events as A intersection B is empty set as well as their
union is S. 24. Two events A and B are independent events if
occurrence or non occurrence of one does not affect the occurrence
or non-occurrence of the other. 25. Example: The event A (get king
card in first attempt) and B(get queen card in second attempt )
when two cards are to be drawn in succession replacing the first
drawn card in pack before second draw. 26. Two events A and B are
dependent events if occurrence or non occurrence of one affect the
occurrence or non-occurrence of the other. 27. Example : The event
A (get king card in first attempt) and B(get queen card in second
attempt ) when two cards are to be drawn in succession not
replacing the first drawn card in pack before second draw 28. There
are three ways to assign a probability, P(Ei), to an outcome, Ei,
namely: Classical approach: make certain assumptions (such as
equally likely, independence) about situation. Relative frequency:
assigning probabilities based on experimentation or historical
data. Subjective approach: Assigning probabilities based on the
assignors judgment. 6.30 29. If an experiment has n possible
outcomes [all equally likely to occur], this method would assign a
probability of 1/n to each outcome. Experiment: Rolling a die
Sample Space: S = {1, 2, 3, 4, 5, 6} Probabilities: Each sample
point has a 1/6 chance of occurring. 6.31 30. If an experiment
results in n mutually exclusive, equally likely and exhaustive
events and if m of these are in favor of the occurrence of an event
A, then probability of event A is written as: P(A)=m / n 31.
Experiment: Rolling 2 die [dice] and summing 2 numbers on top.
Sample Space: S = {2, 3, , 12} Probability Examples: P(2) = 1/36
P(6) = 5/36 P(10) = 3/36 6.33 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7
8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 32.
Relative frequency of occurrence is based on actual observations,
it defines probability as the number of times, the relevant event
occurs, divided by the number of times the experiment is performed
in a large number of trials. 33. Bits & Bytes Computer Shop
tracks the number of desktop computer systems it sells over a month
(30 days): For example, 10 days out of 30 2 desktops were sold.
From this we can construct the 111111111probabilities of an event
(i.e. the # of desktop sold on a given day) 6.35 Desktops Sold # of
Days 0 1 1 2 2 10 3 12 4 5 34. There is a 40% chance Bits &
Bytes will sell 3 desktops on any given day [Based on estimates
obtained from sample of 30 days] 6.36 Deskto ps Sold [X] # of Days
Relative frequency Desktops Sold 0 1 1/30 = .03 .03 =P(X=0) 1 2
2/30 = .07 .07 = P(X=1) 2 10 10/30 = .33 .33 = P(X=2) 3 12 12/30 =
.40 .40 = P(X=3) 4 5 5/30 = .17 .17 = P(X=4) 30 = 1.00 = 1.00 35.
In the subjective approach we define probability as the degree of
belief that we hold in the occurrence of an event P(you drop this
course) P(NASA successfully land a man on the moon) P(girlfriend
says yes when you ask her to marry you) 6.37 36. We study methods
to determine probabilities of events that result from combining
other events in various ways. 6.38 37. There are several types of
combinations and relationships between events: Complement of an
event [everything other than that event] Intersection of two events
[event A and event B] or [A*B] Union of two events [event A or
event B] or [A+B] 6.39 38. Why are some mutual fund managers more
successful than others? One possible factor is where the manager
earned his or her MBA. The following table compares mutual fund
performance against the ranking of the school where the fund
manager earned their MBA: Where do we get these probabilities from?
[population or sample?] 6.40 39. Venn Diagrams 6.41 Mutual fund
outperforms the market Mutual fund doesnt outperform the market Top
20 MBA program .11 .29 Not top 20 MBA program .06 .54 E.g. This is
the probability that a mutual fund outperforms AND the manager was
in a top-20 MBA program; its a joint probability [intersection].
40. Alternatively, we could introduce shorthand notation to
represent the events: A1 = Fund manager graduated from a top-20 MBA
program A2 = Fund manager did not graduate from a top-20 MBA
program B1 = Fund outperforms the market B2 = Fund does not
outperform the market 6.42 B1 B2 A1 .11 .29 A2 .06 .54 E.g. P(A2
and B1) = .06 = the probability a fund outperforms the market and
the manager isnt from a top-20 school. 41. Marginal probabilities
are computed by adding across rows and down columns; that is they
are calculated in the margins of the table: 6.43 B1 B2 P(Ai) A1 .11
.29 .40 A2 .06 .54 .60 P(Bj) .17 .83 1.00 P(B1) = .11 + .06 P(A2) =
.06 + .54 whats the probability a fund outperforms the market?
whats the probability a fund manager isnt from a top school? BOTH
margins must add to 1 (useful error check) 42. Conditional
probability is used to determine how two events are related; that
is, we can determine the probability of one event given the
occurrence of another related event. 6.44 43. Experiment: random
select one student in class. P(randomly selected student is
male/student is on 3rd row) = Conditional probabilities are written
as P(A/ B) and read as the probability of A given B 6.45 44. Again,
the probability of an event given that another event has occurred
is called a conditional probability P( A and B) = P(A)*P(B/A) =
P(B)*P(A/B) both are true 6.46 45. Events: A1 = Fund manager
graduated from a top-20 MBA program A2 = Fund manager did not
graduate from a top-20 MBA program B1 = Fund outperforms the market
B2 = Fund does not outperform the market 6.47 B1 B2 A1 .11 .29 A2
.06 .54 46. Example 2 Whats the probability that a fund will
outperform the market given that the manager graduated from a
top-20 MBA program? Recall: A1 = Fund manager graduated from a
top-20 MBA program A2 = Fund manager did not graduate from a top-20
MBA program B1 = Fund outperforms the market B2 = Fund does not
outperform the market Thus, we want to know what is P(B1 | A1) ?
6.48 47. We want to calculate P(B1 | A1) 6.49 Thus, there is a
27.5% chance that that a fund will outperform the market given that
the manager graduated from a top-20 MBA program. B1 B2 P(Ai) A1 .11
.29 .40 A2 .06 .54 .60 P(Bj) .17 .83 1.00 48. One of the objectives
of calculating conditional probability is to determine whether two
events are related. In particular, we would like to know whether
they are independent, that is, if the probability of one event is
not affected by the occurrence of the other event. Two events A and
B are said to be independent if P(A|B) = P(A) and P(B|A) = P(B)
6.50 49. For example, we saw that P(B1 | A1) = .275 The marginal
probability for B1 is: P(B1) = 0.17 Since P(B1|A1) P(B1), B1 and A1
are not independent events. Stated another way, they are dependent.
That is, the probability of one event (B1) is affected by the
occurrence of the other event (A1). 6.51 50. We introduce three
rules that enable us to calculate the probability of more complex
events from the probability of simpler events The Complement Rule
May be easier to calculate the probability of the complement of an
event and then substract it from 1.0 to get the probability of the
event. P(at least one head when you flip coin 100 times) = 1 P(0
heads when you flip coin 100 times) The Multiplication Rule: P(A*B)
The Addition Rule: P(A+B) 6.52 51. Multiplication Rule for
Independent Events: Let A and B be two independent events, then ( )
( ) ( ).P A B P A P B Examples: Flip a coin twice. What is the
probability of observing two heads? Flip a coin twice. What is the
probability of getting a head and then a tail? A tail and then a
head? One head? Three computers are ordered. If the probability of
getting a working computer is .7, what is the probability that all
three are working ? 52. A graduate statistics course has seven male
and three female students. The professor wants to select two
students at random to help her conduct a research project. What is
the probability that the two students chosen are female? P(F1 * F2)
= ??? Let F1 represent the event that the first student is female
P(F1) = 3/10 = .30 What about the second student? P(F2 /F1) = 2/9 =
.22 P(F1 * F2) = P(F1) * P(F2 /F1) = (.30)*(.22) = 0.066 NOTE: 2
events are NOT independent. 6.54 53. The professor in Example is
unavailable. Her replacement will teach two classes. His style is
to select one student at random and pick on him or her in the
class. What is the probability that the two students chosen are
female? Both classes have 3 female and 7 male students. P(F1 * F2)
= P(F1) * P(F2 /F1) = P(F1) * P(F2) = (3/10) * (3/10) = 9/100 =
0.09 NOTE: 2 events ARE independent. 6.55 54. Addition rule
provides a way to compute the probability of event A or B or both A
and B occurring; i.e. the union of A and B. P(A or B) = P(A + B) =
P(A) + P(B) P(A and B) Why do we subtract the joint probability P(A
and B) from the sum of the probabilities of A and B? 6.56 P(A or B)
= P(A) + P(B) P(A and B) 55. P(A1) = .11 + .29 = .40 P(B1) = .11 +
.06 = .17 By adding P(A) plus P(B) we add P(A and B) twice. To
correct we subtract P(A and B) from P(A) + P(B) 6.57 B1 B2 P(Ai) A1
.11 .29 .40 A2 .06 .54 .60 P(Bj) .17 .83 1.00 P(A1 or B1) = P(A) +
P(B) P(A and B) = .40 + .17 - .11 = .46 B1 A1 56. If and A and B
are mutually exclusive the occurrence of one event makes the other
one impossible. This means that P(A and B) = P(A * B) = 0 The
addition rule for mutually exclusive events is P(A or B) = P(A) +
P(B) Only if A and B are Mutually Exclusive. 6.58 57. In a large
city, two newspapers are published, the Sun and the Post. The
circulation departments report that 22% of the citys households
have a subscription to the Sun and 35% subscribe to the Post. A
survey reveals that 6% of all households subscribe to both
newspapers. What proportion of the citys households subscribe to
either newspaper? That is, what is the probability of selecting a
household at random that subscribes to the Sun or the Post or both?
P(Sun or Post) = P(Sun) + P(Post) P(Sun and Post) = .22 + .35 .06 =
.51 6.59
