1. Welcome to the wonderful world of Probability
1/12/2019
K.RAVEENDRA.Associate Professor & Incharge-Examinations Branch,M.Tech.,MISTE.,MISOI

2. PROBABILITY UNIT-1 [PART A] Experiment Sample space
Discrete and continuous Sample Spaces
Events
Probability –Classical Definition
Probability introduced through Sets and Relative Frequency
Probability Definition and axioms
Mathematical Model of Experiments
Joint Probability
Conditional Probability
Total Probability
Baye’s Theorem
Multiplication Theorem
Independent Events
Permutations & Combinations
1/12/2019
K.RAVEENDRA.Associate Professor & Incharge-Examinations Branch,M.Tech.,MISTE.,MISOI

3. PROBABILITY The Probability can be defined in three ways.
First is the Classical Definition
Second is based on set theory and fundamental axioms.
Third is from the relative Frequency.
1/12/2019
K.RAVEENDRA.Associate Professor & Incharge-Examinations Branch,M.Tech.,MISTE.,MISOI

4. PROBABILITY Experiment Any physical action. Tossing a coin
Throwing a dice
Drawing a card
And many more…..
1/12/2019
K.RAVEENDRA.Associate Professor & Incharge-Examinations Branch,M.Tech.,MISTE.,MISOI

5. PROBABILITY Sample Spaces
The set of all possible outcomes in any given experiment is called the sample space and represented with a symbol S . The sample space is a universal set for the given experiment.
1/12/2019
K.RAVEENDRA.Associate Professor & Incharge-Examinations Branch,M.Tech.,MISTE.,MISOI

6. PROBABILITY Discrete and continuous Sample Spaces Discrete Finite
Tossing a Coin, Throwing a dice,…....
Discrete Infinite
Choosing randomly a positive Integer
Continuous Finite
Obtain a number on Spinning Pointer
Continuous Infinite
Nature (prediction) of random signal
1/12/2019
K.RAVEENDRA.Associate Professor & Incharge-Examinations Branch,M.Tech.,MISTE.,MISOI

7. PROBABILITY Events Draw a Spade Draw a Jack Draw a Queen Draw a Heart
Sub set of sample space.
Gives specific (some) characteristic of experiment.
A Sample space with N elements can have 2N
subsets ( events)
All the operations applicable to sets will apply to events.
Draw a Spade
Draw a Jack
Draw a Queen
Draw a Heart
1/12/2019
K.RAVEENDRA.Associate Professor & Incharge-Examinations Branch,M.Tech.,MISTE.,MISOI

8. Probability - Classical definition
The probability is defined as the ratio of the No.of Favourable Outcomes to the Total No.of Possible Outcomes.
1/12/2019
K.RAVEENDRA.Associate Professor & Incharge-Examinations Branch,M.Tech.,MISTE.,MISOI

9. Probability definition and Axioms
Non-negative Number in sample space.
The probability for the Sample space is always 1.
Union of probability of N no. of. Events is
Summation of N no. of. Events.
Axiom 1:
Axiom 2 : P(S) = 1
Axiom 3:
1/12/2019
K.RAVEENDRA.Associate Professor & Incharge-Examinations Branch,M.Tech.,MISTE.,MISOI

10. Probability as a Relative Frequency
The following leads to the definition of Probability as a Relative Frequency.
(1).Common Sense.
(2).Engineering Observations.
(3). Scientific Observations.
Lim (nH/n) = P(H)
1/12/2019
K.RAVEENDRA.Associate Professor & Incharge-Examinations Branch,M.Tech.,MISTE.,MISOI

11. A real Experiment is defined Mathematically by 3 things.
PROBABILITY
Mathematical Model
A real Experiment is defined Mathematically by 3 things.
(1).Assignment of Sample space.
(2).Definition of Events of Interest.
(3). Making Probability Assignment to the Events.
1/12/2019
K.RAVEENDRA.Associate Professor & Incharge-Examinations Branch,M.Tech.,MISTE.,MISOI

12. For two events A and B, the common elements form the event A B.
PROBABILITY
Joint Probability
For two events A and B, the common elements form the event A B.
The Probability P(A B) is called the joint probability for two events A and B.
For two mutually exclusive events, The joint Probability will be zero i.e. null set.
1/12/2019
K.RAVEENDRA.Associate Professor & Incharge-Examinations Branch,M.Tech.,MISTE.,MISOI

13. PROBABILITY Conditional Probability
Given some Event B with non zero probability P(B)>0,
We define the Conditional Probability of an Event A, given B by P(A/B) =P(A B)/P(B)
P(A/B) saying that the probability of event A may depends on probability of second event B.
1/12/2019
K.RAVEENDRA.Associate Professor & Incharge-Examinations Branch,M.Tech.,MISTE.,MISOI

14. Total Probability Theorem
The probability P(A) of any event A defined on sample space S can be expressed in terms of conditional probabilities if N no. of mutually exclusive events Bn
are given as follows. [where n=1,2,….,N]
P(A)=
1/12/2019
K.RAVEENDRA.Associate Professor & Incharge-Examinations Branch,M.Tech.,MISTE.,MISOI

15. PROBABILITY Baye’s Theorem
1/12/2019
K.RAVEENDRA.Associate Professor & Incharge-Examinations Branch,M.Tech.,MISTE.,MISOI

16. Multiplication Theorem of Probability
It can be used to find the probability of outcomes when an experiment is performing on more than one Event.
If there are N events An n=1,2,….,N, in a given sample space, then the joint probability of all the events can be expressed as follows.
1/12/2019
K.RAVEENDRA.Associate Professor & Incharge-Examinations Branch,M.Tech.,MISTE.,MISOI

17. PROBABILITY Independent Events Mathematically, P(A/B) = P(A) &
We call two events A and B are statistically independent if the probability of occurrence of one event is not affected by the occurrence of other event.
Mathematically, P(A/B) = P(A) &
P(B/A) = P(B)
For statistically Independent Events,
= P(A) P(B)
1/12/2019
K.RAVEENDRA.Associate Professor & Incharge-Examinations Branch,M.Tech.,MISTE.,MISOI

18. PROBABILITY Permutations
For n total elements there are n possible outcomes on the First trial, n-1 on the second, and so forth. For r elements being drawn, the no. of possible sequences of r elements from the original n is denoted by and is given by
where r=1,2,…n
1/12/2019
K.RAVEENDRA.Associate Professor & Incharge-Examinations Branch,M.Tech.,MISTE.,MISOI

19. PROBABILITY Combinations
If the order of sequence is not important, there may be fewer possible sequences of r elements taken from n elements without replacement. The resulting number of sequences where order is not important is called the number of combinations.
where r=1,2,…n
1/12/2019
K.RAVEENDRA.Associate Professor & Incharge-Examinations Branch,M.Tech.,MISTE.,MISOI

20. Random Variable UNIT-1 [PART B] Random Variable
Conditions for a function to be a Random Variable
Classification of Random Variables
Probability Distribution Function
Probability density Function
Real Examples for Distribution and density functions
Gaussian Function
Uniform Function
Exponential Function
Rayleigh Function
Binomial Function
Poisson’s Function
1/12/2019
K.RAVEENDRA.Associate Professor & Incharge-Examinations Branch,M.Tech.,MISTE.,MISOI

21. (Solution): (Problem): A Single card is drawn from a deck of 52-cards.
(a) What is the probability that the card is a jack?
(b) That the card will be a5 or smaller?
(c) That the card is a red 10?
(Solution):
Given a 52-card deck.
(a) Since there are 4 jack cards, The probability of drawing a jack card is
P(jack)=4/52
P(jack)=4/52=1/13
P(jack)=4/52=1/13=
P(jack)=
(b) For drawing a card “5 or smaller”
The card may be 5,4,3,or 2 of spades, clubs, diamonds or hearts
P(5 or smaller) =4 x 4/52
P(5 or smaller) =4 x4/52=16/52
K.RAVEENDRA.Associate Professor & Incharge-Examinations Branch,M.Tech.,MISTE.,MISOI
1/12/2019

22. (Problem): A Single card is drawn from a deck of 52-cards.
(a) What is the probability that the card is a jack?
(b) That the card will be a5 or smaller?
(c) That the card is a red 10?
P(5 or smaller) =4 x4/52=16/52=4/13
P(5 or smaller) =4 x4/52=16/52=4/13 =
P(5 or smaller) =
(c) For drawing a card red 10
Since there are two red cards (one is diamond 10 and another is heart 10)
The required probability is P(red 10)= 2/52
The required probability is P(red 10)= 2/52=1/26
P(red 10)= 2/52=1/26=
P(red 10)=
K.RAVEENDRA.Associate Professor & Incharge-Examinations Branch,M.Tech.,MISTE.,MISOI
1/12/2019