math

1. Introduction to
Probability
Presented by
ASIF RAHAMAN
ROLL – 34900721068
ECE 3RD SEM

2. Introduction
• We often hear such statements: ‘It is likely to rain today
’ , ‘I have a fair chance of getting admission ’ . In each
case , we are not certain of the outcome , but we wish
to assess thple chances of our predictions come true.
• The study of probability provides a mathematical
framework for such assertions and is essential in every
decision making process.

3. Basic Terminology
• Principle of counting
If an event can happen in n1 ways and thereafter for each of these
events a second event can happen in n2 ways , and for each of these
first and second events a third event can happen for n3 ways and so
on , then the number of ways these m event can happen is given by
the product n1.n2.n3…nm .
• Permutations
A permutation of a number of objects is their arrangement in some
definite order.The number of permutation of n different thing taken
r at a time is
n(n-1)(n-2)….(n-r+1) , which is denoted by nPr .

4. Basic Terminology
• Combinations
The number of combinations of n different objects taken r at a
time is denoted by nCr .
nPr = nCr . r!
• Exhastive events
A set of events is said to be exhaustive , if it includes all the
possible events . For example , In tossing a coin there are two
exhaustive cases either head or tail and there is no third possibility.

5. Basic Terminology (cont.)
• Mutually exclusive events
If the occurrence of one of the events procludes the occurrence of
all other then such a set of events is said to be mutually exclusive.
Just as tossing a coin , either head comes up or the tail and both
can’t happen at the same time , i.e., these two mutually exclusive
cases.
• Equally likely events
If one of the events cannot be expected to happen in preference to
another then such events are said to be equally likely.
In tossing a coin , the coming of the head or the tail is equally
likely.

6. Definition of Probability
• If there are n exhaustive, mutually exclusive and equally likely cases of
which m are favourable to an event A , then probability (p) of the
happening of A is
P(A) = m/n
As there are n-m cases in which A will not happen (denoted by A’ ) , the
chance of A not happening is q or P(A’) so that
q= (n-m) /n =1 – m/n = 1- p
P(A’) = 1 – P(A)
P(A)+P(A’) = 1
NOTE
If in n trials , an event A happens m
times , then the probability (p) of
happening of A is given by __
p = P(A) = Lt (m/n)
n→∞

7. Rules of Probability
• Rule 1: 0 ≤ P(A) ≤ 1 for any event A
• Rule 2: The probability of the whole sample space is 1
P(S) = 1
• Rule 3: P(Ac) = 1 – P(A)
• Rule 4: If A and B are disjoint events then
P(A or B) = P(A) + P(B)
• Rule 5: If A and B are independent
P(A and B) = P(A) x P(B)

8. Sample space
• The sample space S of a random process is the set of all possible
outcomes.
• Example : Tossing a coin 3 times
s = { 1,2,3,4,5,6}

9. Reference
• Higher Engineering Mathematics B S Grewal
• Youtube