Probability (Part B)

Probability & Statistics

Experiment: A process carried out under some given conditions and set of rules is termed as
Experiment. e.g. (i) Tossing a die (ii) Conducting an examination of candidates (iii) Observing
rainfall at a given place over a period. (iv) Monitoring traffic on a road etc.
Outcome: The result of an experiment is called outcome
Trail of an experiment: Conducting an experiment and observing the result once is termed as
trail.
Types of Experiments: Based on the nature of result an experiment can be broadly categorized
into two types
Deterministic experiment: A deterministic experiment is one whose outcome is unique under
given conditions and does not vary from trail to trail, e.g. current flowing through a given
conductor.
Random experiment: A random experiment has either a set of outcomes or a range of
outcomes. However on any trail only one outcome (or value) occurs from the given set of
outcomes or range of outcomes. There is always a degree of uncertainty in the outcome of a
random experiment on each and every trail. With the result the outcome of a random experiment
is expected to change from trail to trail.
Sample Space: The set of possible outcomes of a random experiment is called a sample space. It
is denoted by S. It can be discrete (countable) or continuous. Each value or outcome of sample
space is called sample point.
This can be explained by various examples as given below; (See Fig. 1)
(i) Tossing a die. The possible outcomes are {1, 2, 3, 4, 5, 6},
(ii) Marks obtained by a student in an examination of a subject. The possible outcome has range
from 0 to 100.
(iii) Distance to which a shot-put player throws the SHOT. The possible outcome has range 0 to
25m. Where 25 is the maximum range.

1, 2, 3, 4, 5, 6 1, 2, 3, 4 …… 100

Sample space for Die Sample space for Marks of Student

S={x: 0<x<25}

Sample space for shot-put throw of a player

Fig. 1: Sample Space of various examples.

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 Probability (Part B)

Event: On conducting a random experiment an outcome is observed. An event A is said to have

occurred if a particular outcome is observed in a trail of a random experiment, e.g. observing an
odd number once a die is thrown. If odd number is observed one can say that event A has
occurred otherwise event A has not occurred. An event A is a subset of a sample space S. This is
explained in Fig. 2.

A
1, 3, 5,
2, 4, 6
S
Fig.2: Sample space for Die

Probability of an event: It is a function defined over the outcome of an event. It is defined as

nA
P( A) 
N
Where nA denotes number of outcomes favorable to event A and N is total number of possible
outcomes. This is classical definition of probability P(A).
Modern definition of probability is given as
nA
P( A)  Lim N  
N
Where nA denotes number of times event A has occurred when the experiment is conducted large
number of times.
Axioms and Properties of probability
P( A)  0

0  P( A)  1

P( S )  1

If ϕ is a empty set (that is an impossible event), then

P( )  0
If A1, A2, A3, ..., Ak, ..... are mutually disjoint events defined over a sample space S then
 k  k
P  Ak    P( Ai )
 i 0  i 0
If A is an event defined over a sample space and A/ is said to be complement of A then
P( A/ ) 1  P( A)
Two events are said to be non-disjoint if there are some outcomes common to both.
If A and B are two non-disjoint events then
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 Probability (Part B)

P( AUB)  P( A)  P( B)  P( A  B)
Example: A die is thrown. Event A is said to have occurred if an odd number is observed. Event
B is said to have occurred if a number less than 4 occurs. In present case A={1,3,5} B={1,2,3}
A and B are non-disjoint because 1 and 3 are common to both.
Conditional probability:
A and B are two non-disjoint events defined over a sample space S as shown in Fig.3. Assume
that on a certain trial event B occurs. The probability that event A has also occurred when event
B has occurred is known as Conditional probability of the event A given that event B has
occurred. This is denoted as P(A/B) and is given by
P( A  B)
P ( A / B) 
P( B)

S
A A B

Fig.3: Condition Probability of A with respect to B

Total probability theorem: S is a sample space of a random experiment.

A1, A2, A3 , …… An are n disjoint events defined over sample space S as shown in Fig.4.
B is another event defined over S such that B is non-disjoint with events A1, A2 , ……. An
then the probability of B is given by
n
P( B)   P( B / Ak ) P( Ak )
k 1

S
A2 B
A1 A3

Ak
An

Fig. 4: Total probability of B

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 Probability (Part B)

Proof: Since A1, A2, …… An are disjoint events.

Therefore BՈA1, BՈA2, ………… BՈAn are disjoint sets or events.
Hence B= (BՈA1)Ս(BՈA2)………U(BՈAn)
P(B) = P(BՈA1)+ P(BՈA2)+ ………. + P(BՈAn)
n
P( B)   P( B  A )
k 1
k

P( Ak  B)
From conditional probability P( Ak / B) 
P( B)
Therefore P( Ak  B)  P( Ak / B) P( B)
n
Hence P( B)   P( B / Ak ) P( Ak )
k 1

Bayes’ Theorem: Again from Fig.4 S is a sample space and A1, A2, A3, …… An are n
disjoint events defined over S and B is another event defined over S
Then probability of any event Ak conditional to event B is given by
P ( B / Ak ) P ( Ak )
P ( Ak / B )  n
 P( B / Ak )P( Ak )
k 1

This is known as Bayes’ Theorem

P( B  Ak )
Proof: P( B / Ak ) 
P( Ak )
Or P( B  Ak )  P( B / Ak ) P( Ak )
P( B  Ak ) P( B / Ak ) P( Ak )
Now P( Ak / B)  
P( B) P( B)
Applying total probability theorem we get
P ( B / Ak ) P ( Ak )
P ( Ak / B )  n
 P( B / Ak )P( Ak )
k 1

Statistical Independence: Two events A and B are said to be statistically independent if

occurrence of one does not affect the occurrence of other. Mathematically
P( A  B) P( A) P( B)
Bernoulli’s Trials: There are number of random phenomenon in our daily life which have only
two possible outcomes e.g. tossing a coin, result of candidate in an examination, transmission of
a binary bit, firing a missile on a target etc. When such random experiments are conducted
repeatedly these are termed as Bernoulli’s trials.
The two outcomes of a random experiment having only two possible outcomes are designated as
A and A/.
If P(A) = p then P(A/)= 1-p

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 Probability (Part B)

In Bernoulli’s trails the outcomes of successive trials are statistically independent, hence the
probability of observing the event A k number of times in any sequence of N trials is
P(A occurring k times in a sequence) = pk (1-p)N-k
The number of sequences in which event A occurs k times is given by C(N, k). Where C(N, k) is
binomial coefficient and is given by
N!
C(N , k ) 
( N  k )!k!

Thus if an experiment which has only two possible outcomes is conducted N times and the
probability that event A occurs k number of times is given by

P( A occuring k numberof times)  C ( N , k ) p k (1  p) N k

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 Probability (Part B)

Tutorial problems
(i) Write the sample space of following experiments;
(a) Tossing a coin twice.
(b) Tossing two coins simultaneously
(c) Throwing two dice together
(d) Throwing three stones on a target
(e) Result of 4 students in a test
(f) Transmission of a sequence of 4 binary bits.
(ii) Write the sample space associated with the experiment of tossing 3 coins at a time.
If A denotes the event that two ‘H’ appear, write all the sample points of event A.
(iii) A box contains three 10Ω resistors and two 47Ω resistors. Two resistors are drawn
one after another without replacement. Write sample space of this experiment as pairs
of resistors. Also find probabilities of following events (a) both resistors drawn are
10Ω (b) one resistor is 10Ω and another is 47Ω (c) both resistors are 47Ω (d) First
resistor is 47Ω and second is 10Ω.
(iv) Probability that a communication receiver has high selectivity is 0.54 and it has high
fidelity is 0.81 and probability that it has both high is 0.18. Determine the probability
(a) that a system with high fidelity will also have high selectivity and (b) that a
system with high selectivity will also have high fidelity.
(v) In a radar system because of the noise in the system the correct detection is done with
probability p. Assume that p0 is probability that target is present and (1-p0) is
probability that target is not present. Determine the probability that target was
actually present when target was detected.
(vi) Three students appear in a competitive examination. The student will be called for
interview if he secures at least 65 points out of hundred. Draw sample space of this
experiment. What is the probability that (a) all three are called for interview (b) Two
are called for interview (c) None is called for interview.
(vii) Alphanumeric characters are transmitted over a noisy communication channels as 8
bit binary codes. Determine the probability that the character is received correctly if
probability of bit error is 0.1. (b) If one bit error correcting code is employed
determine the probability of recovering the character correctly.
(viii) A communication network divides a long message into 5 packets. The probability that
a packet is received in error is 0.01. The entire message is checked at received end after
assembling it. What is the probability that message is received in error?
(ix) Describe the relationship between methodology and Method.
(x) What do you know by Research? Write down general and specific characteristic fatures
of a good research.
(xi) What are the various methods of acquiring knowledge?
(xii) What are benefits, qualities and objectives of research?
(xiii) Describes the various types of research. Write down the various steps followed while
doing research.

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