Definition of Probability :

1.

Probability is a branch of mathematics that deals with counting the

likelihood of a given event’s occurrence, which is expressed as a value between

1 and 0. An event with a probability of 1 can be considered a sureness event : for example, the probability of a throwing

coin resulting in either “heads” or “tails” is 1, because

there are no other choices, assuming the coin lands flat.

2.

An

event with a probability of 50% can be considered to have equal odds of

occurring or not occurring: for example, the probability of a coin toss resulting

in “heads” is1/2 , because the toss is equally as likely to result in

“tails.” An event with a probability of 0% can be considered an impossible

: for example, the probability that the coin will land (flat) without either

side facing up is 0%, because either “heads” or “tails”

must be facing up. A few paradoxical, probability theory applies precise

calculations to quantify uncertain measures of random events

Probability

Theorems :

Probability

theory is the branch of mathematics concerned with probability. Although there are several different probability

interpretations, probability theory

treats the concept in a rigorous mathematical manner by expressing it through a

set of axioms. Usually these axioms formalise probability in terms of a probability space , termed

the probability measure, to a set of outcomes called the sample space.

*Terminology for probability

theory:

• experiment: operation of observation or

measurement; e.g., coin flip;

•

outcome:

result acquired through an experiment; e.g., coin

shows tails;

•

sample space: set of all possible results of an experiment; e.g., sample space for

coin flip: S = {H, T}. Sample spaces can be finite or infinite

Often we are interested in integration of two or more events. This

can be represented using set theoretic operations . suppose a sample space S and two events A and B:

•

complement A (also A0 ): all components

of S that are not in A;

• subset

A ? B: all components of A are also elements of B;

• union

A ? B: all components of S that are in A or B;

•

intersection A ? B: all components of S that are

in A and B.

*Rules of Probability :

1. Rule of Subtraction: The probability that event A will happen is equal to

1 minus the probability that event A will not happen .

P(A) = 1 – P(A’)

2. Rule of Multiplication : The probability that Events A and B both happen

is equal to the probability that Event A happens times the probability that

Event B happens, given that A has happened.

P(A ? B) =

P(A) P(B|A)

3. Rule of Addition: The probability that Event A or Event B happens

is equal to the probability that Event A happens plus the probability that

Event B happens minus the probability that both Events A and B happen .

P(A ? B) = P(A) + P(B)

– P(A ? B)