The document provides an overview of basic probability concepts, including definitions, laws, and types of events such as independent, dependent, and conditional events. It explains various probability calculations, including a priori and conditional probabilities, and discusses the application of Bayes' theorem. It also covers concepts such as joint and marginal probabilities, as well as the relationships between events in probability.

1. Basic Probability
• Definition & Terminologies
• Basic Laws
• Independent, Dependent,
Conditional
• Bayes theorem
Coverage:

2. Probability
• Probability is simply how likely
something is to happen.
• Many events can't be predicted with total certainty.
The best we can say is how likely they are to happen,
using the idea of probability.
• How likely tossing a coin I will get ‘head’?
• How likely it will rain everyday in the month of
‘shrawan next year?
• How likely you will be a “happy” person in your life?

3. Your turn: give answer
What is the probability that a non-leap
year contains 53 Sundays?
And if it is a leap year?
ANS: 1/7
(& 2/7)
Non-leap yr has 52 full weeks +1 extra day
1 extra day could be Sun, Mon,…Sat
P(extra Sunday) =1/7

4. 58-4
Flipping is an Experiment / Trial
Head + Tail makes Sample Space
Head is Favorable case/Outcome, if we win for Head up
Both Head & Tail are possible outcomes - Exhaustive cases
In one toss both H & T cannot occur - Mutually exclusive
In tossing a fair coin both H & T are Equally Likely
Terminologies
You flip a coin

5. Three type of probability
• a-priori
• empirical
• subjective

6. a-priori (Classical )Probability
E
in
outcomes
of
number
outcomes
of
number
total
:
)
(
e



n
n
N
Where
N
E
P e
Formula is applicable only if:
• Each outcome is equally likely,
exhaustive & mutually exclusive
• Needs a priori -- information of chance

7. Find the probability, if we have such contingency table
• What is the probability that a household is planning to
purchase a large TV? =250/1000
• What is the probability that a household will actually
purchase a large TV? =300/1000
• What is the probability that a household is planning to
purchase a large TV and actually purchases the
television?=200/1000
Purchase behavior of big-screen TV

8. Probability of 2 or more events
2 die: red and blue, are rolled.
What is the probability that we get-
FIVE in blue & SIX in red ?
4-8
Laws for compound events
• Unions and Intersections of Events
• Independent and Dependent Events
• Complementary Events

9. Purchase behavior of big-screen TV
simple
events
Also called
Marginal
events
Joint event:
P(plan and actually purchased) = 200/1000
Complement event: (No plan to purchase),
is complement event of (Plan to purchase)

10. Marginal probability from joint prob
P(A) = P(A & B) + p(A & notB)

11. U = OR, Either X or Y
= AND, both X and Y
X Y
XUY
X Y

12. Mutually Exclusive (disjoint)
• Events with no common elements
are disjoint
Y
X
P X Y
( )
  0 • P(Employed who purchase
TV & unemployed who
purchased TV) = 0
• Male & Female students
• HH in Urban & Rural

13. Independent & Dependent Events
Probability laws differ when
events are independent or when
dependent
#

14. General Law of Addition
)
&
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
Y
X
P
Y
P
X
P
Y
or
X
P
Y
X
P
Y
P
X
P
Y
X
P








Y
X
When events are disjoint
)
(
)
(
)
( Y
P
X
P
Y
X
P 



15. Example: Law of Addition [ X OR Y ]

16. Your turn
Let the events,
N=prefer name brand milk
M= male
Now,
a) P(N)=
660
2276
b) P(M & N) =
319
2276
=.14
c) P(M or N)=
P(M)+P(N)- P(M&N)
=
1138
2276
+
660
2276
−
319
2276
= .6498

17. Break
4-17

18. Independent Events
• Occurrence of one event does not affect the
occurrence or nonoccurrence of the other
event
• Getting H in first toss of coin not affect
getting H again in second toss
• Solving a problem by student X has no
effect on solving same by student Y
• If X & Y are indep. events, P(X∩Y)=0

19. Happening of at least one Independent events
• When events X & Y are two independent
events, happening of at least one:
𝑷 𝑿 ∪ 𝒀 = 𝟏 − 𝑷 𝑿 ∩ 𝒀
= 𝟏 − 𝑷 𝑿 ∩ 𝑷 𝒀
= 𝟏 − 𝑷 𝑿 × 𝑷 𝒀

20. Example: A problem was given to three students X, Y, & Z
whose chances of solving such problem is 0.5, 0.6 & 0.7
respectively.
If all three tries to solve the problem independently, what is
the chance that the problem will be solved?
𝑷 𝑿 ∪ 𝒀 ∪ 𝒁 = 𝟏 − 𝑷 𝑿 ∩ 𝒀 ∩ 𝒁
= 𝟏 − 𝑷 𝑿 ∩ 𝑷 𝒀 ∩ 𝑷 𝒁
= 𝟏 − 𝑷 𝑿 × 𝑷 𝒀 × 𝑷 𝒁
=1- (0.5)x(0.4)x(0.3) = .94
Your turn: if P(A)= 0.2, P(B)=0.3 & P(C )=0.6 for
solving a problem, what is the chance that problem
will be solved?
Ans:
.776

21. Event P or A but not both
4-21 15
.
0
1000
200
2
1000
300
1000
250
)
(
2
)
(
)
(
)
(








X
A
P
P
A
P
P
P
both
not
but
A
OR
P
P
P

22. Neither P /Nor A
4-22
65
.
0
35
.
0
1
)
(
1
)
(
)
(
,
35
.
0
1000
200
1000
300
1000
250
)
(













A
P
P
A
P
P
A
nor
P
said
nether
who
P
so
A
P
P

23. Conditional Probability
Conditional probability of A given B =P(A|B)
)
(
)
(
)
|
(
B
P
B
A
P
B
A
P


4-23
P(not purchased| not planed)
=? 650/750=0.867

24. Joint probability in conditional form
Multiplication rule of probability
)
(
)
(
)
|
(
B
P
B
A
P
B
A
P


)
(
)
|
(
)
( B
xP
B
A
P
B
A
P 


25. Bayes’ Theorem -extended
4-25
  )
(
)
(
)
(
)
(
)
(
)
(
B
P
B
D
P
A
P
A
D
P
A
P
A
D
P
D
A
P


If A & B are me&ex events
& if D is a subset event of {A,B}
the conditional probability
of A given D is:
A B
D
Revised
(posterior)
prior
conditional
joint

26. Example: Suppose two types of drugs(X & Y) is available for curing Degu(D) whose
market share is 40% & 60% respectively.
Also suppose, chances for curing Degu using these drugs is P(D/X)=0.6 &
P(D/Y)=0.7.
One patient visiting the doctor who was suffering from Degu and now cured was
selected randomly, what is the probability that he was using drug X?
Here, P(X, market share)=0.4, P(Y)=.6, Let D= cured from Degu
P(D/X)=0.6 & P(D/Y)=0.7
P(X/D)=P(drug X used | Degu was cured) =
𝐏
𝐃
𝐗
𝐏 𝐗
𝐏
𝐃
𝐗
𝐏 𝐗 +𝐏
𝐃
𝐘
𝐏 𝐘
=
𝟎.𝟔𝐱𝟎.𝟒
𝟎.𝟔𝐱𝟎.𝟒+𝟎.𝟕𝐱𝟎.𝟔
=
.𝟐𝟒
𝟎.𝟐𝟒+𝟎.𝟒𝟐
=
.𝟐𝟒
𝟎.𝟔𝟔
= 𝟎. 𝟑𝟔
Q. What is the probability that he
was taking Drug Y? Ans: 0.64

27. Thanks