the of probability

1. INTRODUCTION
TO
PROBABILITY
SUBJECT CODE : SESH2080
SUBJECT NAME : STATISTICS
FOR MACHINE
LEARNING

2. • REPRESENTING MY TEAM MEMBER’S
ENROLEMENT NUMBERS NAME OF THE
STUDENTS
22SE02ML090 VAMSI
22SE02ML089 PAVAN KUMAR
22SE02ML113 HIMANSHU SHARMA
22SE02ML087 VENU GOPAL
22SE02ML115 AJAY KUMAR

3. INDEX :
1. PROBABILITY INTRODUCTION
2. ASSINGNING PROBABILITY
3. EVENTS AND THEIR PROBABILITY
4. CONDITIONAL PROBABILITY
5. BAYES’ THEOREM

4. Blaise Pascal(1623-
1662) The Father of
Probability

5. PROBABILITY

6. FR
Probability
Probability is a branch of mathematics that deals with calculating
the likelihood of a given event’s occurrence. It is expressed as a
number between 0 and 1.
For calculating probability, we simply divide the number of
favorable outcomes by the total number of outcomes.
Probability Formula
P(E) = Number of favorable outcomes
Total number of outcomes

7. TYPES OF PROBABILITY
Classical. (Also referred to as Theoretical). The
number of outcomes in the sample space is known,
and each outcome is equally likely to occur .
Empirical. (Also referred to as Statistical or
Relative Frequency). The frequency of outcomes is
measured by experimenting .
Subjective. You estimate the probability by making
an "educated guess", or by using your intuition.

8. FR
Probability Formula Examples
Example 1. There are 8 balls in a container, 4 are red, 1
is yellow and 3 are blue. What is the probability of
picking a yellow ball?
solution: The probability is equal to the number
of yellow balls in the
container divided by
the total number of balls
in the container,
i.e. 1/8.

9. Example 2
A page is opened at random from a book containing 200 pages. What is
the probability that the number of the page is a perfect square?
Sol. Let’s assume ,
S be the Sample Space,
A be the event of getting on the page is perfect square.
n(S) = 200
A = { 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196 }
n(A) = 14
P(A) = 14/200 = 7/100
P(A) = 0.07.

10. ASSIGNING
PROBABILITY

11. ASSIGNING PROBABILITY
Basic Requirements for Assigning Probabilities
1. The probability assigned to each experimental outcome must
be between 0 and 1, inclusively.
(0≤P(E) ≤1 for all i)
where:
E; is the with experimental outcome
P(E) is its probability

12. ASSIGNING PROBABILITY
2. The sum of the probabilities for all
experimental outcomes must equal 1.
P(E1) + P(E2) + ... + P(E) = 1
where:
n is the number of experimental outcomes

13. ASSIGNING PROBABILITY
Classical Method :
Appropriate when all outcome are equally likely
P(Outcome) = 1/n , n is number of possible outcomes
Example : Toss a coin – Head , Tail
P(Head) =1/2 ,P(Tail) = ½
Roll a Die – 1, 2, 3, 4, 5, 6
P(1) = 1/6 ,P(2) = 1/6
Relative Frequency Method :
Approrpriate when past data available where experiment has
been repeated many times
P(Outcome) = Proportion of times that the outcome as occur

14. Example : E1 occurred 20 times
E2 occurred 13 times
E1 occurred 17 times
P(E1) = 20/50 0.40
P(E2) = 13/50 0.26
P(E3) = 17/50 0.34
0.40+0.26+0.34 = 1
Subjective Method : Assigning probability based on
judgment
P(E1) = 0.5 P(E2) = 0.4
P(E1)+P(E2) = 0.5+0.4 = 0.9 = 1

15. EVENTS AND
THEIR
PROBABILITY

16. Events and their Probability
Events in probability can be defined as certain
likely outcomes of an experiment that form a
subset of a finite sample space. The probability
of occurrence of any event will always lie
between 0 and 1. There could be many events
associated with one sample space.

17. FR
Independent and Dependent Events
Independent events in probability are those events whose outcome
does not depend on some previous outcome. No matter how many
times an experiment has been conducted the probability of
occurrence of independent events will be the same.
For example, tossing a coin is an independent event in probability .
Dependent events in probability are events whose outcome depends
on a previous outcome. This implies that the probability of
occurrence of a dependent event will be affected by some previous
outcome
For example, drawing two balls one after another from a bag
without replacement

18. Mutually Exclusive Events
Event’s that cannot occur at the same time are known as
mutually exclusive events. Thus, mutually exclusive events in
probability do not have any common outcomes.
For example, S = {10, 9, 8, 7, 6, 5, 4}, A = {4, 6, 7} and B = {10, 9, 8}.
As there is nothing common between sets A and B thus, they are
mutually exclusive events .
Impossible and Sure Events
An event that can never happen is known as an impossible event.
As impossible events in probability will never take place thus, the
chance that they will occur is always 0. For example, the sun revolving
around the earth is an impossible event.A sure event is one that will
always happen. The probability of occurrence of a sure event will
always be 1.
For example, the earth revolving around the sun is a sure event.

19. CONDITIONAL
PROBABILITY

20. Conditional Probability.
Conditional Probability : Let A and B are two events
associated with the same random experiments then
probability of occurred of A under condition B has
already occurred is called Conditional probability .
Formula: P(A|B) = P(A∩B) /P(B),
And P(B|A) = P(A∩B) /P(A)

21. Tossing a Coin:
Let 's consider t wo event s in t ossing
t wo coins be,
A: Getting a head on the first coin.
B: G et t ing a head on t he second coin .
Sample space for t ossing t wo coins is :
S = {HH, HT , T H, T T }
T he conditional probability o f
g et t ing a head on
t he second coin (B) g iven t hat we g ot a
head on t he first coin (A) is = P(B|A )
S i nc e t he coins are independent
(one coin's out come does not affect t he ot her),
P(B|A ) = P(B ) = 0. 5 (50%), which is
t he probabilit y of g et t ing a head on a
sing le coin t oss.

22. Example -2
A Die is rolled .If the outcome is an odd number, What is probability
of that it is prime ?
Solution :
Here,
A=Event that outcome is an odd number.
A={1,3,5}.
∴n(A)=3
Let B=Event that outcome is prime.
Then, B={2,3,5}
∴n(B)=3
Now, (A∩B)={3,5}
⇒n(A∩B)=3
∴ Required probability , P(B/A)=n(A∩B)/n(A)=2/3.

23. Bayes’
Theorem

24. Definition of
Bayes’ Theorem
Bayes' Theorem is a mathematical formula that describes the
probability of an event, based on prior knowledge of conditions that
might be related to the event. It is named after Reverend Thomas
Bayes, who introduced the concept. The formula is expressed as
follows:
P(A∣B)= P(B∣A)⋅P(A)
P(B)

25. Derivation
The derivation of Bayes' Theorem involves using the
definit ion of condit ional probabilit y and t he m ult iplicat ion
rule.
Here's a st ep - by - st ep derivat ion
1)St art wit h t he D efinit ion of Condit ional Probabilit y:
P( A∣ B)=P(A∩ B)/P(B)
2)Apply t he Sym m etry Propert y of Int ersect ion:
P( A∩ B)=P(B∩ A)
3)Use the Multiplication Rule: P(B∩ A)=P(B∣A)⋅P(A)
T he mult iplicat ion rule st at es t hat t he probabilit y
of t he int ersect ion of t wo event s is t he condit ional
probabilit y of one event g iven t he ot her, m ult iplied by t he
probability of the other event.

26. FR
Derivation
4)Substitute into the Conditional Probability Formula: P(A∣B)=
P(B∣A)⋅P(A)/P(B)
Substitute the expression for P(B∩A) into the conditional
probability formula.
P(A∣B)= P(B∣A)⋅P(A)/P(B)
5)This is Bayes' Theorem:
The formula expresses the probability of event A given that event
B has occurred in terms of the conditional probability of B given A,
the prior probability of A, and the marginal probability of B.

27. FR

28. THANK YOU!!!