This document discusses probability theory and its applications. It begins by defining probability as a measure of how likely an event is to occur between 0 and 1. It then provides examples of calculating theoretical probability for simple events like a coin toss or dice roll. The document goes on to explain how probability theory is applied in many areas such as mathematics, statistics, science, and engineering. It provides examples of using probability for risk assessment in fields like finance, biology, and engineering reliability. Finally, it discusses how probability assessments influence decisions and have changed society.

2. Probability is the measure of the likeliness that an event will occur. Probability
is quantified as a number between 0 and 1 (where 0 indicates impossibility and
1 indicates certainty). The higher the probability of an event, the more certain
we are that the event will occur. A simple example is the toss of a fair
(unbiased) coin. Since the two outcomes are equally probable, the probability of
"heads" equals the probability of "tails", so the probability is 1/2 (or 50%)
chance of either "heads" or "tails".
These concepts have been given an axiomatic mathematical formalization
in probability theory (see probability axioms), which is used widely in
such areas of study asmathematics, statistics, finance, gambling, science (in
particular physics), artificial intelligence/machine learning, computer science,
game theory, and philosophy to, for example, draw inferences about the
expected frequency of events. Probability theory is also used to describe the
underlying mechanics and regularities of complex systems.

3. HISTORY OF PROBABILITY
Christiaan Huygens probably published the first book on probability
According to Richard Jeffrey, "Before the middle of the seventeenth century, the
term 'probable' (Latin probabilis) meant approvable, and was applied in that
sense, univocally, to opinion and to action. A probable action or opinion was
one such as sensible people would undertake or hold, in the
circumstances." However, in legal contexts especially, 'probable' could also
apply to propositions for which there was good evidence.
The scientific study of probability is a modern development. Gambling shows
that there has been an interest in quantifying the ideas of probability for
millennia, but exact mathematical descriptions arose much later. There are
reasons of course, for the slow development of the mathematics of probability.
Whereas games of chance provided the impetus for the mathematical study of
probability, fundamental issues are still obscured by the superstitions of
gamblers.

4. Gerolamo Cardano
The sixteenth century Italian polymath Gerolamo
Cardano demonstrated the efficacy of defining odds as the ratio of
favourable to unfavourable outcomes (which implies that the probability
of an event is given by the ratio of favourable outcomes to the total
number of possible outcomes). Aside from the elementary work by
Cardano, the doctrine of probabilities dates to the correspondence
of Pierre de Fermat and Blaise Pascal (1654). Christiaan
Huygens (1657) gave the earliest known scientific treatment of the
subject. Jakob Bernoulli”s Ars Conjectandi (posthumous, 1713)
and Abraham de Moivre's Doctrine of Chances (1718) treatedthe
subject as a branch of mathematics. See Ian Hacking's The Emergence
of Probability and James Franklin's The Science of Conjecture for
histories of the early development of the very concept of mathematical
probability

5.  Probability theory is applied in everyday life in risk assessment and
in trade on financial markets. Governments apply probabilistic
methods in environmental regulation, where it is called pathway
analysis. A good example is the effect of the perceived probability of
any widespread Middle East conflict on oil prices—which have ripple
effects in the economy as a whole. An assessment by a commodity
trader that a war is more likely vs. less likely sends prices up or
down, and signals other traders of that opinion. Accordingly, the
probabilities are neither assessed independently nor necessarily very
rationally. The theory of behavioral finance emerged to describe the
effect of such groupthink on pricing, on policy, and on peace and
conflict.
 In addition to financial assessment, probability can be used to
analyze trends in biology (e.g. disease spread) as well as ecology
(e.g. biological Punnett squares). As with finance, risk assessment
can be used as a statistical tool to calculate the likelihood of
undesirable events occurring and can assist with implementing
protocols to avoid encountering such circumstances.
Applications

6.  The discovery of rigorous methods to assess and
combine probability assessments has changed
society. It is important for most citizens to understand
how probability assessments are made, and how they
contribute to decisions.
 Another significant application of probability theory in
everyday life is reliability. Many consumer products,
such as automobiles and consumer electronics,
use reliability theory in product design to reduce the
probability of failure. Failure probability may influence
a manufacturer's decisions on a product's warranty.
 The cache language model and other statistical
language models that are used in natural language
processing are also examples of applications of
probability theory.

7. PROBABILITY LINE
 Many events can't be predicted with total certainty. The best we can say is
how likely they are to happen, using the idea of probability.
 Tossing a Coin
 When a coin is tossed, there are two possible outcomes:
 heads (H) or
 tails (T)
 We say that the probability of the coin landing H is ½.
 And the probability of the coin landing T is ½.
 Throwing Dice
 When a single die is thrown, there are six possible outcomes: 1, 2, 3, 4, 5, 6.
 The probability of any one of them is 1/6.
 Probability
 In general:
 Probability of an event happening =Number of ways it can happen
Total number of outcomes

8. Probability does not tell us exactly
what will happen, it is just a guide

9. What is Theoretical Probability?
While probability theory focuses on the likelihood of an event taking place,
theoretical probability is all about the occurrence of an event based on all
possible outcomes that are already known. To simply explain this statement, take
for instance the tossing of a coin. We know that the result would be either head or
tail, which are equally likely. By means of theoretical probability, we know that
the likelihood of a head or tail coming on top is same i.e. ½ or 0.5.
Most real-life events can be predicted based on their occurrences in the past by
studying the data collected from surveys, trials, opinion polls etc. Such kind of
probability theory is called Empirical Probability and it’s based on observations
and experiences. For example, if we have to find the probability that a child will
choose vanilla ice cream from a range of flavors, then we need the data from a
survey. Suppose from a group of 100 children, 35 chose vanilla, then the
probability that a child will prefer vanilla flavor over others is 35/100 i.e. 0.35.

10.  Theoretical probability on the other hand uses the knowledge on
the likely outcomes to find the different ways in which the event
can occur. It is the method of finding the probability of an event
from a sample space of known and equally likely outcomes. This
simple means that each outcome is as likely to occur as the
other. To explain this with an example, take for instance the
rolling of a fair die. How do you find the probability of rolling a
‘six’? You first need to determine the sample space of equally
likely events. In this case, it’s (1, 2, 3, 4, 5, and 6). The
theoretical probability of rolling a six is 1 out of 6, i.e. 1/6. The
probability of rolling six will be represented as P(E), where P is
the theoretical probability and E is the event in consideration.
 Therefore, we can say that theoretical probability uses analytical
knowledge on the probable outcomes to determine the
probability of an event instead of using experimentation.

11. PREDICTABLE AND UNPREDICTABLE
OCCURRENCES
 Predictable Occurrences: The time an object takes to hit the
ground from a certain height can easily be predicted using
simple physics. The position of asteroids in three years from
now can also be predicted using advanced technology.
 Unpredictable Occurrences: Not everything in life, however,
can be predicted using science and technology. For
example, a toss of a coin may result in either a head or a
tail. Also, the sex of a new-born baby may turn out to be
male or female. In these cases, the individual outcomes are
uncertain. With experience and enough repetition, however,
a regular pattern of outcomes can be seen (by which certain
predictions can be made).

14. IMAGES