student education

1. Basic Probability

2. Basic Concepts
• Probability –Chance , Likelihood ,likely , odds
• Probability can be defined as the measure of
the likelihood that a particular event will occur
• Numerical measure is between 0 and 1
• Probability theory provides with mechanism
for measuring and analyzing uncertainties
associated with future events
P = Favorable Outcomes / Total Outcomes

3. Basic Terms
1.Experiment- Any activity that generates data
Experiment has two properties
a) Each experiment has several possible
outcomes and all these outcomes are known in
advance
b) None of these outcomes can be predicted
with certainty
Ex toss of a coin, rolling a dice etc

4. Experiments
• Experiment Possible Outcomes
1.Tossing a fair coin Head ,Tail
2. Rolling a dice 1,2,3,4,5,6
3.Selection of item Good ,Bad
4.Introduce new product Success ,Failure

5. Basic Terms cont..
2 .Event –outcome or a set of outcomes of an
activity or result of a trial.
Ex getting 2 heads in the trial of tossing three
fair coins simultaneously would be an event
3. Elementary Event – simple event , is a single
possible outcome of an experiment
Ex toss of coin, event of head coming up is an
elementary event , denoted as P(A) where A is
the event

6. Basic Terms cont…
4. Joint Event –compound event , combination
of 2 or more elementary event
Ex drawing a black ace from a pack of cards , it
has 2 elementary events – ‘black’ and ‘ace’
5.Simple Probability –simple or elementary
event occurs
Ex drawing a ‘diamond’ card from pack of 52
cards is a simple event
Here P(A)=13/52 or 1/4

7. Basic Terms cont…
6. Joint Probability – occurrence of 2 or more
simple events
Ex drawing a ‘black ace ‘ from a pack of cards.
This is 2 simple events
- card being ‘black’
- card being ‘ace’
P(Black Ace)or P(A) = 2/52 , as there are 2 ‘black
aces’ in the pack

8. Basic Terms Cont…
7.Sample Space – Collection of all possible
events or outcomes of an experiment
Ex 2 possible outcomes of toss of a coin
Sample Space S = (H,T)
P(S)=P(H,T)=1
Ex Roll of a dice
S=(1,2,3,4,5,6)

9. Basic Probability Relationships
• Complement of an Event – Compliment of any
event A is the collection of outcomes that are
not contained in A, denoted as A’
• Outcomes contained in A and outcomes
contained in A’ must equal sample space S
P(A)+P(A’)=P(S)=1
P(A) = 1-P(A’)

10. Basic Probability Relationships
• Mutually Exclusive Events
Two events are said to be mutually exclusive , if
both events cannot occur at the same time as
outcome of single experiment
Ex toss of a coin – either ‘Head’ or ‘Tail’ will
occur, not both. Hence these are mutually
exclusive events

11. Mutually Exclusive Events
Addition Rule
It 2 events A and B are mutually exclusive ,
Probability that either of them will occur is the sum
of their separate probabilities
Ex roll of dice , Probability of getting either 5 or 6
P(A or B) = P(A) + P(B)
P(5 or 6)= P(5) + P(6)
= 1/6+1/6 = 1/3
P(A or B) is also written as P (A U B) or P (A Union B)

12. Non Mutually Exclusive Events
• If A and B are non mutually exclusive events
P(A U B)= P(A) + P(B) –P(A and B)
Or P(A U B)= P(A) + P(B) –P(A and B)
P(A and B) = P (A intersection B) or P(AB)
Events (A and B) consists of all those events which are contained in both A and B simultaneously
Ex Pack of cards
P(A) = An Ace is drawn
P(B) = A spade is drawn
P(AB) = An “Ace Spade” is drawn
P (A U B) = P(A)+P(B)-P(AB)
= 4/52+13/52-1/52
= 4/13
Logic : Subtract P(AB) because Spade is counted twice –once in event A (4 Aces) and once again in event
B (13 cards of spade including the ace)

13. VENN DIAGRAMS

16. Numerical- Non mutually exclusive
• Survey of 100 persons revealed that 50 persons read “India
Today” , 30 persons read “Time” , and 10 of those 100 read
both “India Today” and “Time”
• P(A) =50
• P(B)=30
• P(AB)=10
• P(A U B) = person reads “India Today” or “Time” or both
• P(A U B) = P(A) +P(B) –P(AB)
= 50/100 + 30/100 -10/100
=70/100
= 0.7

17. Independent Events
• Two events A and B are said to be “Independent
Events” if the occurrence of one is not influenced
at all by the occurrence of other
Ex if 2 fair coins are tossed , then the result of one
toss is totally independent of the result of the other
toss .The Probability that ‘head’ will be outcome of
any one toss will always be ½ ,irrespective of
whatever the outcome is of the other toss . Hence
these 2 events are independent

18. Multiplication Rule
• Applied to compute the Probability if both events A and B occur at the
same time .The multiplication rule is different if two events are
independent as against the two events being non independent
• If 2 events are independent , Probability that both will occur is
P(AB)= P(A) * P(B)
Ex Toss of coin “Twice” , Probability that 1st toss is “head” and second toss is
‘tail’ is
P(HT) = P(H) * P (T)
P(HT) = 1/2*1/2=1/4
If A and B are not independent , ie Probability of occurrence of an event is
dependent or conditional upon occurrence or non occurrence of the other
event , it is Conditional Probability

19. Numericals
1. Tow dices are tossed .Elaborate the sample
space (Outcomes)
(1,1),(1,2),(1,3)……….(1,6)
(2,1),……………………….(2,6)
(3,1),………………………(3,6)
(4,1),………………………(4,6)
(5,1),………………………(5,6)
(6,1),……………………….(6,6)
Total outcomes =36

20. Numercials
1.What is Probability of getting sum of 5 in two
throws of dice
2. Probability of sum of 8
3.Probability of sum of 8 or more ( 8,9,10,11,12)

26. Problem
Q) A class consist of 100 students , 25 are girls ,
75 are boys . 20 are rich and remaining poor , 40
are fair complexioned
What is the probability of selecting a fair
complexioned rich girl?
(Hint : Fair , Rich , Girl)

27. Solution
• Probability of fair person = 40/100=2/5
• Probability of rich person =20/100 =1/5
• Probability of girl =25/100=1/4
• Since all events are independent, using
multiplication rule
• Probability of fair & rich girl=2/5*1/5*1/4
= 2/100= 0.02

28. Problem
Q) A candidate is selected for interviews for
three posts .For the first post , there were 3
candidates , for the second post 4 and for the
third 2
What is the probability that the candidate is
selected for at least one post?
(Hint: Compliment of an Event)

29. Solution

30. Dependent or Contingent Events
• Events are said to be dependent if the
occurrence of any one event affects the
occurrence of the other events

31. Dependent Event

32. Dependent Event cont..

33. Problem
Q) Two boxes contain respectively 6 brown , 8
blue , 1 black ball AND 3 brown , 7 blue and 5
black balls
One ball is drawn from each box , what is the
probability that both the balls drawn are of the
same color

34. Solution
Probability that both are brown = 6/15*3/15=18/225
Probability that both are blue =8/15*7/15=56/225
Probability that both are black = 1/15*5/15= 5/225
Since these events of drawing both the balls of same color are
mutually exclusive
Probability = 18/225 + 56/225 + 5/225=79/225
(Since if you draw 2 brown balls , then you cant draw 2 blue balls
and so on or if you draw 2 black balls , you cant draw 2 blue balls
etc)

35. Problem
• A company is interested in understanding the consumer behaviour of
the capital of the newly formed state Chhattisgarh, that is Raipur. For this
purpose, the company has selected a sample of 300 consumers and asked
a simple question, “Do you enjoy shopping?” Out of 300 respondents, 200
were males and 100 were females. Out of 200 males, 120 responded
“Yes,” and out of 100 females, 70 responded “Yes.”
• A respondent is selected at random. Construct a probability matrix and
ascertain the probability that:
• the respondent is a male
• enjoys shopping
• is a female and enjoys shopping
• is a male and does not enjoy shopping
• is a female or enjoys shopping
• is a male or does not enjoy shopping
• is a male or female

36. Solution

37. Solution

38. Problem
• ABRC, a leading marketing research firm in India, wants to collect
information about households with computers and Internet access
in urban Mumbai. After conducting an intensive survey, it was
revealed that 60% of the households have computers with Internet
access; 70% of the households have two or more computer sets.
Suppose 50% of the households have computers with Internet
connection and two or more computers. A household with
computer is randomly selected.
• What is the probability that the household has computers with
Internet access or two or more computers?
• What is the probability that the household has computers with
Internet access or two or more computers, but not both?
• What is the probability that the household has neither computers
with Internet access nor two or more computers?

39. Solution

40. Naïve Bayes Theorem-Conditional
Probability
P (A/B) = P (B/A) * P(A)/P(B)
P ( Queen Diamond) = P (Q/D)
P(Queen/Diamond) = P(D/Q)*P(Q)/P(D)
P (Q) = 4/52 = 1/13
P (D) = 13/52=1/4
P (D/Q)= 1/4
P( Q/D)= 1/4*1/13 / ¼
P(Q/D) =1/13

41. Problem
• Roll of a dice and we know that the number
that came up is larger than 4
• What is the probability that the outcome is an
even number , given that it is larger than 4

42. Solution
Let P(A) = Even Number
P(B) = Larger than 4
P(A/B) = P(B/A) *P(A)/P(B)
P(A) = 3/6 = 1/3
P(B) = 2/6 = 1/3
P(B/A)= Larger than 4 Given its Even = 1/6 ( ie 6 )
P(A/B) = 1/6*1/3 / 1/3
P(A/B) =1/6 ( ie number 6)