Superior University
Lahore
Sahiwal Campus
|
|
|
|
Department:
|
|
|
Programme:
|
B.S(I.T)
|
|
Course:
|
Statistics and Probability
|
|
Topic:
|
Subjective
Probability, Addition Law for M.E Events, Simple & Compound Events
|
|
Submitted to:
|
M Zeeshan sarwar
|
|
Submitted by:
|
Prof. Fyaz hussain
|
|
Roll no:
|
028
|
|
Date of Submission:
|
16/11/2015
|
(shanich108@gmail.com)
Subjective Probability
DEFINITION
OF 'SUBJECTIVE PROBABILITY'
A probability derived from an
individual's personal judgment about whether a specific outcome is likely to
occur. Subjective probabilities contain no formal calculations and only reflect
the subject's opinions and past experience.
BREAKING
DOWN 'SUBJECTIVE PROBABILITY'
Subjective probabilities differ from
person to person. Because the probability is subjective, it contains a high
degree of personal bias. An example of subjective probability could be asking
New York Yankees fans, before the baseball season starts, the chances of New
York winning the World Series. While there is no absolute mathematical proof
behind the answer to the example, fans might still reply in actual percentage
terms, such as the Yankees having a 25% chance of winning the world series.
(Read more: Subjective Probability Definition | Investopedia http://www.investopedia.com/terms/s/subjective_probability.asp#ixzz3rWsATVJu )
(Read more: Subjective Probability Definition | Investopedia http://www.investopedia.com/terms/s/subjective_probability.asp#ixzz3rWsATVJu )
Addition Law for Mutually Exclusive Events
First go
through the mutually exclusive events. These are the two or more than two
events that have nothing in common. Theorem for Addition law of mutually
exclusive events states that:
If there
are two events say “x” and “y” and both are mutually exclusive events, then the
probability that either “x” or “y” occur is the sum of the probabilities of
both the events.
Suppose
there are two events named as “x” and “y”, both events have nothing in common.
Addition of such event is done by following the addition law for mutually
exclusive events.
P(x
or y) = p(x u y) = p(x) + p(y)
Let make it clear by solving
simple examples.
EXAMPLE 1:
A single
card is selected from a deck of 52 cards. Find the probability that the
randomly selected card is either king or queen.
Solution:
Total
number of outcomes = 52
Probability
of each single card = 1/52
Now let
first find the probability of queen
Probability
of queen is represented by = P(Q)
There are
four queens in a deck of playing cards.
Therefore,
probability of queen = P(Q) = 4/52
Now, let
find the probability of king
Probability
of king is represented by = P(k)
In the
same manner there are four kings in a deck of 52 playing cards.
Therefore,
probability of queen = P(K) = 4/52
Addition
law of mutually exclusive event is used, as the requirement is to find the
probability of king or queen.
Therefore,
P(K or Q) = P(K u Q) = P(K) + P(Q) = 4/52 + 4/52 = 8/52
EXAMPLE 2:
A dice is
thrown. Find the probability that the face is less than three or it is multiple
of 5.
Solution:
When a
dice is rolled, there are six possible outcomes
Sample
space = S = {1, 2, 3, 4, 5, 6}
Let first
find the probability of face that are less than 3.
A = {1, 2}
= 2/6
Now second
part is that the face is multiple of 5.
B = {5} =
1/6
Use
addition law, as again there is nothing in common between the two events.
P (A or B)
= P(A u B) = P(A) + P(B) = {1, 2} + {5} = {1, 2, 5}
P(A or B) = 2/6 + 1/6 = 1/2
Definition Of Equally Likely
The
outcomes of an experiment are equally likely to occur when the probability of
each outcome is equal.
Example of Equally Likely:
·
When you toss a fair coin, you are equally likely to get a head
or a tail.
·
When you roll a fair die, you are equally likely to roll a 1, 2,
3, 4, 5, or 6.
Ques: Which one of the following best replaces the
blank? 'A and B are playing tennis. The event of A or B winning the match is
______________.'
Choices:
A. a likely event
B. an unlikely event
C. an equally likely event
D. none of the above
Correct Answer: C
B. an unlikely event
C. an equally likely event
D. none of the above
Correct Answer: C
Solution:
Step 1: Both A and B
have equal chances of winning the match.
Step 2: So, it is an equally likely event.
Step 2: So, it is an equally likely event.
EVENTS
Event is
the subset of the sample space or event can also be defined as the collection
of either one or more than one outcomes of an experiment. It is sum of all
possible outcomes of an experiment. Sample space forms after combining all the
events. Any part of the sample space can be named as an event. Event may
be of one outcome or a combination of more than one outcome.
Event with
a single outcome is named as simple event and an event with having two or more
than two outcomes is known as compound event.
SIMPLE EVENTS:
Simple
events can be defined as the single outcome of the performed experiment or it
is an event which cannot be broken down any more.
COMPOUND EVENTS:
Compound
events are the combination of two or more than two simple event. It can also be
defined as an event that contains more than one sample points in it.
EVENT EXAMPLES:
1.
If a single face is considered when a die is rolled, then it
will be simple event. For example suppose getting 5 or 6 or 3 or 2 etc… on the
die when it is thrown, is called as simple event. If the event is any even
number on the die, then the event is consist of points {2, 4, 6}, which is
known as compound event. That compound event is consisting of three simple
events i.e., {2}, {4} and {6}.
2.
Suppose two dice are rolled simultaneously, then the pair (1, 1)
will be the simple event. This is so, because it is a single outcome in the
sample space. If event consists of the sum of two dice is ‘’5’’ than it
consists of four outcomes i.e., (1, 4), (2, 3), (3, 2), (4, 1) and this is
considered to be a compound event.
3.
Suppose two coins are tossed simultaneously, then the pair (HT)
will be the simple event. If condition is defined that an event should consist
of at least one head then there are three outcomes. These outcomes are (HH),
(HT) and (TH) and this is said to be as compound event. That compound event
consists of three simple events i.e., {HH}, {HT} and {TH}.
NUMERICALS...
Q#1:
A Marble drawn At Random From a box containing 10 red, 30 white, 20 blue and 15 orange marbles. Find the probability that is:
Ø Orange or Red
Ø Not Red or Blue
Ø Not Blue
Ø White
Ø Red, White or Blue
SOLUTION:
RED=10 P
(RED) =10/75
WHITE=30 P
(WHITE) =30/75
BLUE=20 P
(BLUE) =20/75
ORANGE=15 P
(ORANGE) =15/75
TOTAL=75
Ø
Orange Or Red
P (Orange Or Red) = P (Orange)+ P(Red)
15/75+10/75=25/75=1/3
Ø
Not Red Or Blue
P (Not Red Or Blue) = P (White) +P (Orange)
30/75+15/75=45/75=3/5
Ø
Not Blue
P (Not Blue)= P(White)+P(Red)+P(Orange)
30/75+10/75+15/75+15/75=11/15
Ø
White
P (White) =30/75
Ø
Red, White Or Blue
P (Red, White Or Blue) =P(Red)+P(White)+P(Blue)
10/75+30/75+20/75=60/75
Q#2
Of 12 eggs in a refrigerator, 2 are bad. From these, 4 eggs are chosen at random to make cake. What are probabilities that:
Ø Exactly one is bad
Ø At least one is bad
SOLUTION:
Total eggs= 12
Bad eggs= 2
Good eggs= 10
n=4
P (exactly 1 bad egg) = 2C1 10C3/12C4=0.48
P (at least one is bad) = 2C1 10C3 + 2C2 10C2/12C4 =0.575
Q#3
A bag contains 14 identical balls, 4 of which are red, 5 black and 5 white. Six balls are drawn from the bag. Find the probability that
Ø 3 are red
Ø At least two are white
SOLUTION:
N = 14
Red = 4
Black = 5
White = 5
n = 6
P (three are red) = 4C3+ 10C3/14C6= 124/3003=0.041
P (at least two are white) = 5C2. 9C4+ 5C3.9C3+ 5C4. 9C2+
5C5.9C1/ 14C6= 2289/3003=0.7
