Download M2L4 Probability of Events

M2L4
Probability of Events
1. Introduction
In this lecture, details of various concepts related to probability of events, such as, equality of
events, concept of field, countable and non-countable space, conditional probability, total
probability, Bayes’ Theorem etc. is discussed.
2. Equality of Events
2.1.
Equal Events
Any two events ‘ ’ and ‘ ’ are called equal if they consist of same elements.
2.2.
Events with Equal Probability
Events ‘ ’ and ‘ ’ are called equal with probability 1, if the set consisting all outcomes those
are in ‘ ’ or in ‘ ’ but not in
, has zero probability. It is mathematically denoted as,
Thus the events ‘ ’ and ‘ ’ are equal with probability 1 if and only if
.
It must be noted here that if only
, then ‘ ’ and ‘ ’ are equal in probability, but
no conclusion can be drawn about the probability of
; ‘ ’ and ‘ ’ might be mutually
exclusive.
2.3.
Field
A field is defined as a nonempty subset of events, called a class of events, in such a way that:
then
. Or if
and
then
.
if
and
Other properties of a field are: if
i.
, also
ii.
and
Since
then
and
is not an empty set and contains at least one event ‘ ’, also contain
and
and hence
.
3. Countable Spaces
If the sample space, ‘ ’ contains ‘ ’ outcomes and ‘ ’ is a finite number, then the
probabilities of all outcomes can be expressed in terms of the probabilities of the elementary
events
. However, it should follow the Axioms, i.e.,
,
.
If ‘ ’ is an event having ‘ ’ elementary elements
elementary events
, ‘ ’ can be written as union of the
. Thus,
.
This is also true even if a set ‘ ’ comprised of an infinite but countable number of elements
,
…
4. Noncountable Spaces
If ‘ ’ is the set of all real numbers, its subsets might be considered as sets of points on the
real line. It is generally impossible to assign probabilities for all subsets of ‘ ’ to satisfy the
axioms. This is true for any n-dimensional space. Probability space on the real line can be
constructed considering all events at intervals
, and their countable unions and
intersections. In this case, probability is assigned to the event
. Probability of all
other events can be determined with the help of the axioms.
5. Assignment of Probability
The probability of an event
,
can be interpreted as a mass. If a sample space, ‘ ’
and
consists of a finite number of outcomes
given by
are elementary events
, then sum of probability of all events will be
. Mathematically,
. Venn diagram is shown in Fig. 1.
Fig. 1. Assignment of Probability
6. Conditional Probability
If ‘ ’ and ‘ ’ are two events such that
, the probability of ‘ ’ given that ‘ ’ has
already occurred, is obtained as the ratio of probability of intersection of ‘ ’ and ‘ ’, and
probability of ‘ ’. It is mathematically expressed as,
(1)
Venn diagram is give below in Fig. 2.
Fig. 2. Conditional Probability
For any three events
probability of
the probability that all of them occur is the same as the
times probability of
given that both
and
given that
has occurred times the probability of
have occurred. It is mathematically expressed as –
(2)
The Venn diagram is shown in Fig. 3.
Fig. 3. Extended Conditional Probability
This theorem can be generalized for any n number of events (
).
7. Total Probability
Sometimes occurrence of one event, ‘ ’, cannot be determined directly. It depends on
occurrence of other events,
which are mutually exclusive and collectively
exhaustive (union is the entire sample space, ). Then to calculate the probability of ‘ ’, a
weightage of the probabilities of events
are used. This approach is known as
Theorem of Total Probability.
Theorem of Total Probability: If any event ‘ ’ must results in one of the mutually exclusive
and collectively exhaustive events,
, then probability of ‘ ’ can be
mathematically derived as:
(3)
The explanatory Venn diagram is given in Fig. 4.
Fig. 4. Venn diagram for Total Probability Theorem
Problem 1. Municipality of a city uses
of its required water from a nearby river and
remaining from the groundwater. There could be various reasons of not getting sufficient
water from the sources including pump failure, non-availability of sufficient water and so on.
If probability of shortage of water due to the system involve with river is
and that with
groundwater is
, what is the probability of insufficient supply of water to the city?
Solution: Let us first denote the events mentioned in the problem,
Event A: insufficient supply of water to the city
Event R: water from the river
Event G: water from groundwater
Thus we get,
Using
,
Theorem
of
,
Total
,
Probability,
8. Independence
Independent Event: If probability of an event ‘ ’ occurring completely unaffected by
occurrence of event ‘ ’, then event ‘ ’ and ‘ ’ are said to be independent. Thus, if ‘A’ and
‘B’ are independent, it can be expressed as:
or equivalent as,
.
9. Bayes’ Theorem/Rule
If
are mutually exclusive and collectively exhaustive events, then for any event,
for probability of
, given that an event ‘ ’ comprising
has
already occurred, can be denote as,
(4)
Proof: From Conditional Probability theorem, we get,
.
Again, from Joint Probability theorem,
,
Thus
Again from Total Probability theorem,
So, finally
For
practical
problems, Bayes’ Theorem, the probability of different states,
, are generally obtained from the belief of an engineer based on
previous experiences. These probabilities are known as prior, whereas,
is known as
are known as likelihood, which are obtained
posterior. Similarly, the probabilities
from previous experiments. The denominator, the total probability can be treated as a
constant. Thus, the probability can be treated as a constant. Thus, the Bayes’ rule is also
expressed as:
(5)
Problem 2. A particular construction material is ordered from three different companies.
Company ‘ ’ delivers
units per day, out of which
do not satisfy the specific quality.
Company ‘ ’ delivers
units per day, out of which
do not satisfy the specific quality.
Company ‘ ’ delivers
units per day, out of which
do not satisfy the specific quality.
a) What is the probability that one unit of the material picked at random will not satisfy the
specific quality? b) If a load is found to be substandard, what is the probability that it came
from supplier ‘ ’?
Solution.
a) The substandard unit may come either from company ‘ ’ or ‘ ’ or ‘ ’. Thus theorem total
probability should be applied to obtain the probability event , that is selecting a substandard
unit at random.
b) Once it is known that the unit is substandard, the probability of unit being supplied by a
particular company is not same as that when the information of substandard unit was not
known. Thus using Bayes’ theorem to calculate the probability:
Problem 3. Design of foundation for tall structure needs to know the depth of soil above
bedrock, denoted as . Four categories of ‘ ’ are denoted as
;
;
;
. Belief of the geologist states
that the prior probabilities for these four events are as follows:
and
,
,
. A seismic recorder is being used to measure . The
performance of the instrument is shown in the following table:
The readings are obtained at a site and found to be m for the 1st reading (sample#1) and m
for the 2nd reading (sample#2). Calculate the probability of different events (
etc.)
given that the record obtained from successive readings. (from Kottegoda and Roso, 2008)
Solution. The sample#1 was found to be at m, which corresponds to
. The
posterior
probabilities of actual state are obtained from Bayes’ Theorem as follows:
Now,
Thus,
It can be noticed that probability of actual state to be
to the availability of sample 2. However, still about
is increased to
from
due
chance is there that it may not be
. Thus another sample is collected, which is m.
So, Then,
0.675.
Thus,
So, it is noticed that after obtaining sample, the chance of true state being
is very high
. Thus with the help of Bayes’ theorem the probability of unknown are improved
with availability of more information.
10. Concluding Remarks
Probability of events is discussed with different theorems and numerical problems. The
theorem of conditional probability computes the probability of one event based on occurrence
of other event(s). Lastly, the concepts of Total Probability Theorem and Bayes’ Theorem are
useful for revising or updating probability of one event with the availability of more
information. The concept of Random Variables will be discussed in the next lecture.