Counting Principles, Sample space and Events, Operation on Events, Statement of theorems on Probability

1. MIT Arts, Commerce and Science College,
Alandi, Pune
DEPARTMENT OF STATISTICS
F.Y.B.Sc.(Comp. Sci.)
Mathematical Statistics
Unit 1 – Theory of Probability
Prof. Archana Kadam.

2. Techniques for counting:
In many real-life problems you want to count the
number of possibilities. Some basic counting
techniques which will be useful in determining the
number of different ways of arranging or selecting
objects. The two basic counting principles are
given below:
1) Addition Principle
2) Multiplication Principle

3. The Additive Rule of Counting:
Suppose we carry out k operations, but we can not do at the
same time.
Let
n1 = the number of ways the first operation can be performed
n2 = the number of ways the second operation can be
performed
.
.
ni = the number of ways the ith operation can be performed
Then N = n1 + n2 + ….+nk = the number of ways the k
operations can be performed in sequence.

4. Example : A man travel from Pune to Mumbai.
Solution: Number of ways he can travel from Pune to
Mumbai ______

5. Example : A person want to purchase a TABLET or
MOBILE.
Solution: Number of ways he can buy either TABLET
or MOBILE _______

6. Example : Rohan wants to go to the cinema to watch
movie.
Solution : : Number of ways he can watch movie_____

7. The Multiplicative Rule of Counting:
Suppose we carry out k operations in sequence
Let
n1 = the number of ways the first operation can be performed
n2 = the number of ways the second operation can be performed
once the first operation has been completed.
.
.
.
ni = the number of ways the ith operation can be performed once
the first (i - 1) operations have been completed. i = 2, 3, … , k
Then N = n1n2 … nk = the number of ways the k operations can
be performed in sequence

9. Example: A man is checks his wardrobe for something
to wear
Solution: A man can arrange an outfit________

10. Example: A person is going to buy car.
Solution: Number of ways he can buy Car_______

11. Permutation and Combination

12. Permutation is an ordered arrangement of objects.
1. Permutation of ‘r’ object out of ‘n’ (r ≤ n) distinct
object without repetition.

13. 2. Permutation of ‘r’ object out of ‘n’ (r ≤ n) distinct object
with repetition.
n x n x n x ……x n = nr
3. Permutation of ‘n’ object not all distinct:
The number of permutations of n objects of which n1 are of one
kind, n2 are of second kind, ..., nk are of kth kind and the rest if
any, are of different kinds is given as,

14. Combination is an unordered arrangement of objects.
Combination of ‘r’ object out of ‘n’ (0 ≤r ≤ n).

15. Remark on Combination:

16. Deterministic Experiment:
The experiment which have only one possible
outcome i.e. who's result is certain and unique is called
Deterministic Experiment.
OR
The result of experiment is predictable with
certainty and is known to prior to its conduct.
Examples:
1. Heating water above 100 degree C
2. Throwing stone in sky
3. V= u + at

17. Non-Deterministic Experiment Or Random
Experiments:
The experiment which have more than one
possible outcome, all outcomes are known in
advanced but can not predict outcome before
conducting experiment is called Non-Deterministic
Experiment.
Examples:
1. Tossing a Coin
2. Rolling a Die

18. Sample Space
A set of All possible outcome of random experiment is
called sample space. It is denoted by “S” or “Ω”.
Examples:
1. Tossing a Coin outcomes S = { Head, Tail}

19. Types of Sample Space
1)Discrete Sample Space:
If an elements in sample space are finite or countably infinite then it is
called Discrete sample space.
Example:
1) Experiment of Rolling a Die
Finite S = {1,2,3,4,5,6}
2) Number of Seeds germinated out of 10 planted seeds
S = { 0,1,2,3,…….10}
3) Tossing a coin Till Head Appear
Countably S = {H, TH, TTH, TTTH, …………}
Infinite 4) Number of Customers entering in to Departmental Store
S = { 0,1,2,3,4,5,……..}

20. 2) Continuous sample space:
If an elements in sample space are uncountably infinite then it is called
continuous sample space.
Example:
1) Measure the lifetime of a given computer memory chip in a specified
environment. S = { 0 : ∞}

21. Event:
Any subset of sample space is called event. It is denoted by capital letters
A, B, E, F, ....
Example 1) Experiment of Rolling a Die, the S = {1,2,3,4,5,6}
Then Say
Event A shows even number occurs on a face of a die i,.e. A = { 2,4,6};
Event B shows Odd number occurs on a face of a die i,.e. B = { 1,3,5}
Event C shows perfect square occurs on a face of a die i,.e. C = { 1,4}
Event D shows number on a face of a die is divisible by 5 i,.e. D = { 5 }
And Event E = {1,2,3,4,5,6}. We can also Defined F = { }=ф.

22. Types of Event
1) Sure Event:
An event which contain all the elements of sample space is called Sure event.
Example : In a Experiment of Rolling a Die, Event E = {1, 2,3,4,5,6}
Then Event E called Sure Event
2) Elementary Event:
An event which contain only one elements of sample space is called elementary
event.
Example : In a Experiment of Rolling a Die, Event D = { 5 }
Then Event D called elementary Event
3) Impossible Event:
An event which do not contain any elements of sample space or an empty set
is called impossible event.
Example : In a Experiment of Rolling a Die, Event F = { } = ф
Then Event F called impossible Event

23. Set operations on Events
1. Union of Two events
Let A and B be two events, then the union of A
and B is the event (denoted by AB) defined by:
A  B = {e| e belongs to A or e belongs to B}
A  B
A B

24. 2. Intersection of two events
Let A and B be two events, then the intersection
of A and B is the event (denoted by AB) defined
by:
A  B = {e| e belongs to A and e belongs to B}
A  B
A B

25. 3. Complement of an Event
Let A be any event, then the complement of A
(denoted by ) defined by:
= {e| e does not belongs to A}
A
A
A
A

26. In problems you will recognize that you are
working with:
1. Union if you see the word or,
2. Intersection if you see the word and,
3. Complement if you see the word not.

27. 4. Mutually Exclusive Event
Two events A and B are called mutually exclusive if:
They have no outcomes in common.
They can’t occur at the same time. The outcome of the random
experiment can not belong to both A and B
A B 
 
A B

28. 5. Exhaustive Event:
Let A and B are any two events defined on sample
space are said to be exhaustive event if union of both
event represent a sample space. i.e. AUB = S.

29. Definition: Probability of an Event E.
Suppose that the sample space S = {e1, e2, e3, … eN}
has a ‘N’ finite number of outcomes.
Also each of the outcomes is equally likely.
Then for any event E
 
 
 
  no. of outcomes in
=
total no. of outcomes
n E n E E
P E
n S N
 
 
: the symbol = no. of elements of
n A A
Note

30. Probability Model:
Let Ω = { x1, x2, x3, ……xn} be a finite sample space for a
random experiment. A probability model is constructed
by assigning to each sample point of Ω, a real no P(xi)
such that:
1. P(xi) ≥ 0 for all ‘i’
2. ∑ P(xi) = 1
Axioms of Probability:
Let event A defined on “S”. Then
1. P(A) ≥ 0
2. ∑ P(S) = 1
3. P(AUB) = P(A)+P(B), Where A and B are mutually
exclusive event.

31. Theorems on Probability

32. 1. For any event A defined on ”S” 0 ≤P(A) ≤1.
2. Let A’ be the compliment of an event A, then P(A)+P(A’) = 1
3. The probability of an impossible event is zero i.e. P(ф) = 0.
4. If A is the subset of B then P(A) ≤ P(B).
5. Addition theorem: If A and B are any two events defined on “S”,
then P(AUB) = P(A)+P(B)-P(AПB).
6. If A and B are any two events defined on “S”, then
P(A’П B) = P(B) – P(A П B) and P(A П B’) = P(A) – P(A П B).
7. De’Morgans Law:
P(A’ U B’) = P (AПB)’ & P (A’ П B’) = P(A U B)’