1.1 What is probability ?
Probability is a branch of mathematics that deals with the study of uncertainty and randomness. It provides a framework for quantifying uncertainty and making predictions in situations where outcomes are not certain.
Probability theory is widely applied in various fields such as statistics, finance, science, and engineering to model and analyze uncertain situations and make informed decisions based on the likelihood of different outcomes.
There are two models to describe the phenomena in the real world : deterministic models and probabilistic models. These two different types of mathematical models used in various fields to represent and analyze systems or phenomena.
(1) Deterministic Model:
In a deterministic model, the outcome is completely determined by the initial conditions and the parameters of the model.
There is no randomness or uncertainty involved in the system being modeled.
Given the same set of initial conditions and parameters, a deterministic model will always produce the same result.
Deterministic models are often used when the system being studied is assumed to be predictable and without random variations.
Example: A simple mathematical equation like \(y=mx+b\) represents a deterministic model. If you know the values of \(m\), \(x\), and \(b\), you can precisely determine the value of \(y\).
(2) Probabilistic Model:
In a probabilistic model, the outcome is described using probability distributions, reflecting the inherent uncertainty or randomness in the system.
These models acknowledge that certain events or parameters are subject to variation and are not known with certainty.
Probabilistic models are commonly used when dealing with systems involving random or unpredictable elements.
They provide a range of possible outcomes and the likelihood of each outcome occurring.
Example: A model predicting the likelihood of rain tomorrow could be probabilistic. It may provide a probability distribution of possible rainfall amounts, rather than a single deterministic prediction.
In summary, deterministic models assume perfect predictability and do not account for randomness, while probabilistic models incorporate uncertainty and provide a range of possible outcomes with associated probabilities. The choice between these models depends on the nature of the system being studied and the degree of certainty or randomness involved in its behavior.
We can’t predict outcomes perfectly in probabilistic models. However, different outcomes can follow a certain pattern. This is what we discussed in probability theory.