1. Basic Probability
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A RAJASEKHAR YADAV

2. Introduction Probability is the study of randomness and uncertainty.
In the early days, probability was associated with games of chance (gambling).

3. Simple Games Involving Probability
Game: A fair die is rolled. If the result is 2, 3, or 4, you win $1; if it is 5, you win $2; but if it is 1 or 6, you lose $3.
Should you play this game?

4. Random Experiment
a random experiment is a process whose outcome is uncertain.
Examples:
Tossing a coin once or several times
Picking a card or cards from a deck
Measuring temperature of patients
...

5. Events & Sample Spaces Sample Space Event Simple Events
The sample space is the set of all possible outcomes.
Event
An event is any collection
of one or more simple events
Simple Events
The individual outcomes are called simple events.

6. Example Experiment: Toss a coin 3 times. Sample space 
 = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}.
Examples of events include
A = {HHH, HHT,HTH, THH}
= {at least two heads}
B = {HTT, THT,TTH}
= {exactly two tails.}

7. Basic Concepts (from Set Theory)
The union of two events A and B, A  B, is the event consisting of all outcomes that are either in A or in B or in both events.
The complement of an event A, Ac, is the set of all outcomes in  that are not in A.
The intersection of two events A and B, A  B, is the event consisting of all outcomes that are in both events.
When two events A and B have no outcomes in common, they are said to be mutually exclusive, or disjoint, events.

8. Example
Experiment: toss a coin 10 times and the number of heads is observed.
Let A = { 0, 2, 4, 6, 8, 10}.
B = { 1, 3, 5, 7, 9}, C = {0, 1, 2, 3, 4, 5}.
A  B= {0, 1, …, 10} = .
A  B contains no outcomes. So A and B are mutually exclusive.
Cc = {6, 7, 8, 9, 10}, A  C = {0, 2, 4}.

9. Rules Commutative Laws: Associative Laws: Distributive Laws:
A  B = B  A, A  B = B  A
Associative Laws:
(A  B)  C = A  (B  C )
(A  B)  C = A  (B  C) .
Distributive Laws:
(A  B)  C = (A  C)  (B  C)
(A  B)  C = (A  C)  (B  C)

10. Venn Diagram  A B
A∩B

11. Probability
A Probability is a number assigned to each subset (events) of a sample space .
Probability distributions satisfy the following rules:

12. Axioms of Probability For any event A, 0  P(A)  1. P() =1.
If A1, A2, … An is a partition of A, then
P(A) = P(A1) + P(A2) P(An)
(A1, A2, … An is called a partition of A if A1  A2  … An = A and A1, A2, … An are mutually exclusive.)

13. Properties of Probability
For any event A, P(Ac) = 1 - P(A).
If A  B, then P(A)  P(B).
For any two events A and B,
P(A  B) = P(A) + P(B) - P(A  B).
For three events, A, B, and C,
P(ABC) = P(A) + P(B) + P(C) -
P(AB) - P(AC) - P(BC) + P(AB C).

14. Example
In a certain population, 10% of the people are rich, 5% are famous, and 3% are both rich and famous. A person is randomly selected from this population. What is the chance that the person is
not rich?
rich but not famous?
either rich or famous?

15. Intuitive Development (agrees with axioms)
Intuitively, the probability of an event a could be defined as:
Where N(a) is the number that event a happens in n trials

16. Here We Go Again: Not So Basic Probability

17. More Formal: W is the Sample Space: w in W is a single outcome
Contains all possible outcomes of an experiment
w in W is a single outcome
A in W is a set of outcomes of interest

18. Independence P(A,B,C) = P(A)P(B)P(C)
The probability of independent events A, B and C is given by:
P(A,B,C) = P(A)P(B)P(C)
A and B are independent, if knowing that A has happened
does not say anything about B happening

19. Bayes Theorem
Provides a way to convert a-priori probabilities to a-posteriori probabilities:

20. Conditional Probability
One of the most useful concepts! W A B

21. Bayes Theorem
Provides a way to convert a-priori probabilities to a-posteriori probabilities:

22. Using Partitions:
If events Ai are mutually exclusive and partition W

23. Random Variables
A (scalar) random variable X is a function that maps the outcome of a random event into real scalar values W
X(w) w

24. Random Variables Distributions
Cumulative Probability Distribution (CDF):
Probability Density Function (PDF):

25. Random Distributions:
From the two previous equations:

26. Uniform Distribution
A R.V. X that is uniformly distributed between x1 and x2 has density function: X1 X2

27. Gaussian (Normal) Distribution
A R.V. X that is normally distributed has density function: m