Probability is a way of expressing
knowledge or belief that an event will occur or has occurred. The concept has an exact mathematical meaning in probability theory,
which is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, artificial intelligence/machine
learning and philosophy to draw conclusions about the
likelihood of potential events and the underlying mechanics of complex systems.
Mathematical
treatment
Consider an experiment that can produce a number of results. The
collection of all results is called the sample space of the experiment. The power set of the sample space is
formed by considering all different collections of possible results. For
example, rolling a die can produce six possible results. One collection of
possible results gives an odd number on the die. Thus, the subset {1, 3, 5} is
an element of the power set of the sample space of die rolls. These collections are called
"events." In this case, {1, 3, 5} is the event that the die falls on
some odd number. If the results that actually occur fall in a given event, the
event is said to have occurred.
A probability is a way of assigning every event a value between zero and one, with the requirement
that the event made up of all possible results. (In our example, the event {1,
2, 3, 4, 5, 6} is assigned a value of one.) To qualify as a probability, the
assignment of values must satisfy the requirement that if you look at a
collection of mutually exclusive events (events with no common results, e.g.,
the events {1, 6}, {3}, and {2, 4} are all mutually exclusive), the probability
that at least one of the events will occur is given by the sum of the
probabilities of all the individual events.
The probability of an event A is written as P(A),
p(A) or Pr(A).This mathematical definition of probability can
extend to infinite sample spaces, and even uncountable sample spaces, using the
concept of a measure.
http://en.wikipedia.org/wiki/Probability
Frequency
probability is the interpretation of probability that defines an event's probability as the limit of its relative frequency in a
large number of trials.
Definition
Frequentists talk about probabilities only when dealing with
well-defined random experiments. The set of all possible outcomes
of a random experiment is called the sample space of the experiment. An event is defined as a particular subset of the sample space that
you want to consider. For any event only one of two possibilities can happen;
it occurs or it does not occur. The relative frequency of occurrence of an
event, in a number of repetitions of the experiment, is a measure of the probability of that event.
Thus, if n_{t} is the total number of trials and n_{x} is the number of trials where the event x occurred, the probability P(x) of the event occurring will be approximated by the relative
frequency as follows:
.
(Our solved example in mathguru.com
uses this concept)
http://en.wikipedia.org/wiki/Frequency_probability
The above explanation is copied from
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