Chapter1_Probability , Statistics Reliability

Dr.E.Prasad
Asst Professor, Dept of Mathematics
VIT Bhopal University
Spring 2024-25
Probability Theory
MAT3003 Probability, Statistics and Reliability
Module 1 Dr.E.Prasad

Module-1;Probability Theory
Probability, Statistics and Reliability
Dr.E.Prasad
Syllabus
Introduction to probability concepts, Random experiments, Events,
Conditional probability, Independent events, Theorem of Total
Probability, Baye’s theorem Sample space.

Dr.E.Prasad
Definition1:Random Experiment
A random experiment is an experiment in which
a) All outcomes of the experiment are known in advance
b) Any performance of the experiment results in an outcome, which is not
known in advance
c) The experiment can be repeated any no. of times under identical conditions
A particular performance of the experiment is called a trial and the possible outcomes
are called events or case

Dr.E.Prasad
Definition2: Outcome: A result of a random experiment.
Definition3: Sample Space: The set of all possible out comes, denoted by S
Definition4: Event: A subset of the sample space, denoted by E
Examples:
1. Random experiment: toss a coin; sample space: S={heads, tails} or as we usually
write it, {H,T},events are {H} and{T}
2. Random experiment: roll a die; sample space: S={1,2,3,4,5,6}
3. Random experiment: observe the number of iPhones sold by an Apple store in
Boston in 20152015; sample space: S={0,1,2,3,⋯}
4. Random experiment: observe the number of goals in a soccer match; sample
space: S={0,1,2,3,⋯}
5. Random experiment: Two coins are tossed S={HH, HT, TH, TT} and events are
E1 ={HH} , E2 ={TT}, E3 ={TH}, E4 ={HT}

Dr.E.Prasad
Types of Events
▪ Exhaustive Events
▪ Favorable Events
▪ Mutually Exclusive Events
▪ Equally Likely Events
▪ Independent Events
▪ Dependent Events

Dr.E.Prasad
Definition5: Exhaustive Events : The total number of possible outcomes in any
random experiment is known as exhaustive events (or) exhaustive cases.
Examples:
1. Random experiment: Two coins are tossed S={HH, HT, TH, TT}
Then the number of exhaustive events = 4
2. Random experiment: roll a die; sample space: S={1,2,3,4,5,6},
Then the number of exhaustive events = 6

Dr.E.Prasad
Definition6: Favorable Events :
The number of cases favorable to an event in a trial is the number of outcomes which
entail the happening of the event.
Examples
1. Random experiment: Two coins are tossed S={HH, HT, TH, TT}
Let E be the event selecting two heads, then the number of favorable cases = 1
Let E be the event selecting exactly one head, # of favorable cases = 2

Dr.E.Prasad
**Definition7: Mutually Exclusive Events: Events are said to be mutually
exclusive (or) incompatible if the happening of any one of the events excludes (or)
precludes the happening of all the others i.e.) if no two or more of the events can
happen simultaneously in the same trial. (i.e.) The joint occurrence is not possible.
I,e,.. Two events E1 , E2 are said to be mutually exclusive iff E1∩ E2 = ∅
1. If a coin is tossed either head will turn up or tail will turn up. Both head and tail
cannot turn up simultaneously. So we say that the events head turn up and tail turn
up are mutually exclusive.
If E1 = an event selecting prime number,={2,3,5} E2 =an event selecting even
number ,{2,4,6}, E1∩ E2 ≠ {2},Therefore E1, E2 not mutually exclusive
Examples

Dr.E.Prasad
Definition8: Equally Likely Events: Outcomes of a trial are said to be equally
likely if taking in to consideration all the relevant evidences, there is no reason to
expect one in preference to the others. (i.e.) Two or more events are said to be equally
likely if each one of them has an equal chance of occurring.
I,e,.. Two events E1 , E2 are said to be Equally likely iff E1∪ E2 = S
Examples:
If E1 = an event selecting odd numbers,={1,3,5}, E2 =an event selecting even
numbers ,{2,4,6}, E1∪ E2 = 𝑆,Therefore E1, E2 are equally likely events

Dr.E.Prasad
Definition9: Independent Events : Several events are said to be independent if
the happening of an event is not affected by the happening of one or more events.
Definition10: Dependent Events : If the happening of one event is affected by
the happening of one or more events, then the events are called dependent events.
Note: In the case of independent (or) dependent events, the joint occurrence is possible
**It will be discussed at the time of conditional probability

Dr.E.Prasad
**Definition10: Probability:
If an experiment results in ‘n’ exhaustive cases which are mutually exclusive and
equally likely cases out of which ‘m’ events are favorable to the happening of an
event ‘E’, then the probability ‘p’ of happening of ‘E’ is denoted by P(E) and
defined by
P( E )=
Favorable cases of an event E
Exhaustive cases
=
Favorable cases of an event E
Total number of cases
=
m
n
Axioms of Probability

Dr.E.Prasad
1.Two dice are tossed. What is the probability of getting (i) Sum 6 (ii) Sum 9? Ans:5/36,1/9
2. A card is drawn from a pack of cards. What is a probability of getting (i) a king (ii) a spade (iii)
a red card (iv) a numbered card? Ans:4/52,13/52,26/52,36/52
3. What is the probability of getting 53 Sundays when a leap year selected at random? Ans:2/7
4. A class consists of 6 girls and 10 boys. If a committee of three is chosen at random from the
class, find the probability that (i) three boys are selected, and (ii) exactly two girls are selected?
Ans:3/14 ,15/56
5. A and B throw alternately a pair of dice. A wins if he throws 6 before B throws 7 and B wins if
he throws 7 before A throws 6. Find their respective chance of winning, if A begins. then find the
probability that A wins. Ans 30/61
6. A, B and C in order toss a coin. the first one to throw a head wins. then a)find the probability
that A wins, b)find the probability that B wins c) find the probability that C wins ; Ans 4/7,2/7,1/7
Exercise

Dr.E.Prasad
Theorems of Probability
Let A and B be any two events which are not mutually exclusive
1. P (A or B) = P (A∪B) = P (A + B) = P (A) + P (B) – P (A∩B)
2. P (A and B) = P (A ∩ B) = P (AB) = P (A) + P (B) – P (A ∪ B)
In case of 3 events A,B and C
Let A ,B and C be any three events which are not mutually exclusive
P (A∪B∪C) = P (A) + P (B) + P (C) – P (A∩B) – P (B∩C) – P (A∩C) + P (A∩B∩C)
1.Addition Theorem on Probability
*2.Multiplication Theorem on Probability
If A and B be any two events which are not independent, then (i.e.) dependent.
P (A and B) = P (A∩B) = P (AB) = P (A). P (B/A)
= P (B). P (A/B)
Where P (B/A) and P (A/B) are the conditional probability of B given A and A given B
respectively.
Note:
1. If A and B be any two events which are independents P (A and B) = P (A∩B)= P (A) P(B)
2. If A and B be any two events which are mutually exclusive events P (A∩B) = 0

Dr.E.Prasad
Proof of Addition Theorem on Probability

Dr.E.Prasad
Basic Probability Set theory formula
1. P(φ) = 0
2. σi Pi=1
3. P(Ac
) = 1 − P A
4. P(A ∩ Bc
)= P(A-B) =P(A∩ B) − P(B)
5. P(Ac
∩ Bc
) = P(( AUB)c
)=1- P(A U B)
6. P(Ac
U B)= P(Ac
)+ P(A ∩ B)= 1 − P A + P(A ∩ B)
7. P(Ac
U Bc
) = P ( A ∩ B)c
=1- P(A ∩ B)
8. P(
A
Bc)=
P(A∩Bc)
P(Bc)

Dr.E.Prasad
1. From a pack of 52 cards one card is drawn at random, and then find the probability that it is
either a king or a queen.(Ans 2/13)
2. From a pack of 52 cards one card is drawn at random, and then find the probability that it is
either a spade or an ace.(Ans 4/13)
3. 3 students A,B and C are in a swimming race. A and B have the same probability of winning
and each is twice as likely to win as C. Find the probability that B or C wins. (Ans 3/5)
4. Probability of solving specific independently by A and B are 1/2 and 1/3 respectively. If both
try to solve the problem independently, find the probability that
(i) the problem is solved (Ans 2/3)
(ii) exactly one of them solves the problem.(Ans 1/2)
5. A problem in Mathematics is given to 4 students. A, B, C and D. Their chances of solving the
problems respectively are 1/3,1/4,1/5 and 2/3 is independent . What is the probability that (i)
the problem will be solved ? (ii) at most one of them will solve the problem ?(Ans
13/15,49/90)
6. A, B, C are aiming to shoot a balloon independently, A will succeed 4 times out
of 5 attempts. The chance of B to shoot the ballon is 3 out of 4 and that of C is 2 out of 3. If
the three aim the ballon simultaneously, then find the probability that atleast two of them hit
the balloon.(Ans 5/6)
7. Suppose A and B are independent events with P(A)=0.6, P(B)=0.7 compute (i) P(A ∩ B) (ii)
P(AUB) (iii) P(B/A) (iv) P(Ac
∩ Bc
) v) P(Ac
U B) [Ans. i)0.42 ii)0.88 iii)0.7 iv)0.12 v)?]
Exercise

Dr.E.Prasad
Definition11: Conditional Probability:
Conditional probability is the probability that depends on a previous result or event.
With the help of conditional probability, we can tell apart dependent and independent
events. When the probability of one event happening doesn’t influence the probability of
any other event, then events are called independent, otherwise dependent events.
:
Let’s consider two events A and B, then the formula for conditional probability of A
when B has already occurred is given by:
P(A|B) = P (A ∩ B) / P(B)
https://probability.oer.math.uconn.edu/wp-
content/uploads/sites/2187/2018/01/prob3160ch4.pdf

Dr.E.Prasad
1. Suppose you roll two dice. What is the probability that the sum is 8 given that the first die
shows a 3?(Ans 1/6)
2. A bag contains 3 red and 2 black marbles. Two are drawn at random, the first not being
replaced before the second draw. What is the probability that 2 red marbles are drawn?(Ans
3/10)
3. Two dice are rolled. Consider the events A = {sum of two dice equals 3}, B = {sum of two dice
equals 7 }, and C = {at least one of the dice shows a 1}. (a) What is P (A | C)? (b) What is P (B |
C)? (c) Are A and C independent? What about B and C?
(Ans P(A/C)=2/11,P(B/C)=2/11 and both cases are not independent)
4. Raghava wants to take either a Biology course or a Chemistry course. His adviser estimates that
the probability of scoring an A in Biology is 4/ 5 , while the probability of scoring an A in
Chemistry is 1/ 7 . If Raghava decides randomly, by a coin toss, which course to take, what is his
probability of scoring an A in Chemistry?(Ans.1/14)
5. A family has 2 children. Given that one of the children is a boy, what is the probability that the
other child is also a boy?(Ans.1/3)
6. An urn contains 10 black and 5 white balls. Two balls are drawn from the urn one after the other
without replacement. What is the probability that both drawn balls are black? (Ans 3/7)
7. Raghava is 80% sure he forgot his textbook either at the Union or in Monteith. He is 40% sure
that the book is at the union, and 40% sure that it is in Monteith. Given that Raghava already
went to Monteith and noticed his textbook is not there, what is the probability that it is at the
Union?(Ans.2/3)
Exercise

Dr.E.Prasad
Total Probability Theorem
Statement :Let events C1 , C2 . . . Cn form partitions of the sample space S, where all the events
have a non-zero probability of occurrence. For any event, A associated with S, according to the
total probability
P(A) = σ𝑖=0
𝑛
𝑃(𝐶𝑖)𝑃(𝐴|𝐶𝑖)

Dr.E.Prasad
Proof:

Dr.E.Prasad
Example: A person has undertaken a mining job. The probabilities of completion of job on
time with and without rain are 0.42 and 0.90 respectively. If the probability that it will rain is
0.45, then determine the probability that the mining job will be completed on time

Dr.E.Prasad
Example: Each of Rohit’s three bags holds 100 marbles.
Bag 1 contains 75 red and 25 blue marbles
Bag 2 contains 40 blue and 60 red marbles.
Bag 3 contains 45 red and 55 blue marbles.
Rohit randomly selects one of the bags and then randomly selects a marble from the selected
bag. How likely is it that the selected marble is red? (Ans 0.60)

Dr.E.Prasad
1. If 40% of boys opted for maths and 60% of girls opted for maths, then what is the
probability that maths is chosen if half of the class’s population is girls?(Ans 0.5)
2. Company A produces 10% defective products, Company B produces 20% defective products
and C produces 5% defective products. If choosing a company is an equally likely event,
then find the probability that the product chosen is defective.(Ans 0.116)
3. Suppose 5 men out of 100 men and 10 women out of 250 women are colour blind, then find
the total probability of colour blind people. (Assume that both men and women are in equal
numbers.(Ans 0.045)
4. The probability that the political party A does a particular work is 30% and the political party
B doing the same work is 40%. Then find the probability that the work is completed if the
probability of choosing the political party A is 40% and that of B is 60%. .(Ans 0.36)
5. In 1989 there were three candidates for the position principal- Mr. Chatterji, Mr. Ayangar
and Dr. Sing, whose chances getting the appointment are in the proportion 4:2:3
respectively. The probability that Mr.Chatterji selected would introduce co-education in the
college is 0·3. The probabilities Mr. Ayangar and Dr. Singh doing the same are respectively
0.5 and 0·8. What is the probability that there was co-education in the college in 199O?
(Ans 23/45)
Exercise

Dr.E.Prasad
1. https://www.sanfoundry.com/probability-statistics-questions-answers-theorem-total-
probability/#google_vignette
2. https://objectstorage.ap-mumbai-
1.oraclecloud.com/n/bmzytd5z5pt3/b/Class12/o/1653055057-ncert-0.pdf
Reference

Dr.E.Prasad
Statement: Let S be the sample space and E1,E2, E3 , … , Enbe n mutually exclusive and
exhaustive events with P(Ei) ≠0; i = 1, 2, .., n. Let A be any event which is a sub-set of
E1 ∪ E2∪ E3, … ,∪ En (i.e. at least one of the events E1, E2, …, En ) with P(A) > 0 [Notice that
up to this line the statement is same as that of law of total probability], then
𝑃
𝐸𝑖
𝐴
=
𝑃(𝐸𝑖)𝑃
𝐴
𝐸𝑖
σ𝑖=0
𝑛
𝑃(𝐸𝑖)𝑃
𝐴
𝐸𝑖
*Baye's Theorem
https://egyankosh.ac.in/bitstream/123456789/20490/1/Unit-4.pdf

Dr.E.Prasad
1. The chances of defective screws in three boxes A, B, C are 1/5, 1/6, 1/7respectively. A box
is selected at random and a screw drawn from it at random from it is found defective. The
probability that it came from the box A.(Ans=42/107)
2. Consider two urns. The first contains two white and seven black balls, and the second
contains five white and six black balls. We flip a fair coin and then draw a ball from the
first urn or the second urn depending on whether the outcome was heads or tails. What is
the conditional probability that the outcome of the toss was heads given that a white ball
was selected?(Ans=22/67)
3. A laboratory blood test is 95 percent effective in detecting a certain disease when it is, in
fact, present. However, the test also yields a “false positive” result for 1 percent of the
healthy persons tested. (That is, if a healthy person is tested, then, with probability 0.01,
the test result will imply he has the disease.) If 0.5 percent of the population actually has
the disease, what is the probability a person has the disease given that his test result is
positive? (Ans=0.323)
Exercise

Dr.E.Prasad
5. The content in an urns I, II and III are allows:
1 white, 2 black and 3 red balls,
2 white, 1 black and 1 red balls, and
4 white, 5 black and 3 red balls, One urn is chosen at random and two balls drawn,
they happen to be white and red. What is the probability .that they come urns I?
(Ans 33/118)
6. In a bolt factory machines A, B and C manufacture respectively 25%.35% and 40% of the
total. Of their output 5, 4, 2.percent is defective bolts. A bolt is drawn at random the
product and is found to be defective. What are the probabilities that it was
manufactured by machines C?(Ans.16/69)
7. A doctor is called to see a sick child. The doctor has prior information that 90% of sick
children in that neighborhood have the flu, while the other 10% are sick with measles.
Let F stand for an event of a child being sick with flu and M stand for an event of a child
being sick with measles. Assume for simplicity that F ∪ M = Ω, i.e., that there no other
maladies in that neighborhood. A well-known symptom of measles is a rash (the event of
having which we denote R). Assume that the probability of having a rash if one has
measles is P(R | M) = 0.95. However, occasionally children with flu also develop rash, and
the probability of having a rash if one has flu is P(R | F) = 0.08. Upon examining the child,
the doctor finds a rash. What is the probability that the child has measles?(Ans.0.57)

Dr.E.Prasad

Chapter1_Probability , Statistics Reliability

More Related Content

Similar to Chapter1_Probability , Statistics Reliability (20)

Recently uploaded (20)

Chapter1_Probability , Statistics Reliability