Basic Concepts of Probability & Probability Distributions

Probability is defined as a chance or likelihood of happening of an event. It is basically degree of likelihood that something will happen. Probability measures the likelihood that something specific will occur. For example, a tossed coin has an equal chance, or probability, of landing with one side up ("heads") or the other ("tails"). Probability uses numbers to explain chance. If something is absolutely going to happen, its probability of occurring is 1, or 100 percent. If something absolutely will not happen, its probability of occurring is 0, or 0 percent. The probability of an event is a number lying in the interval 0≤p≤1, with 0 corresponding to an event that never occurs and 1 to an event that is certain to occur. For an experiment with N equally likely outcomes the probability of an event A is n/N, where n is the number of outcomes in which the event A occurs

This course comprises of:

Basic concepts of Probability using illustrative examples

Basic Theorems of Probability

Probability of happening of atleast one event

Baye's Theorem in Probability applying Joint probability and conditional probability

Probability Distributions

Calculation of Probability when there are various possible events using Binomial Distribution

Poisson Distribution using mean number of cases

Normal Distribution & Normal Distribution curves

Importance of Probability:

· The probabilistic understanding of biological processes such as genetic inheritance, evolution, and epidemics, has been essential for scientific progress . The recent explosion in the amount of data from genome projects and other sources, such as microarray experiments, has led to the need for new probability models to understand both the structure of the data and the underlying biology.

· The application of probability to finance has revolutionized an industry. Without the probabilistic models that provide reliable pricing of derivative securities and guide the management of associated risk, these markets could not exist.

· Markov chain Monte Carlo methods allow the investigation of stationary distributions of complex probabilistic structures which are used in analyzing complex day to day problems.

· In computer science, randomized algorithms using probability enable the solution of complex problems that would otherwise be inaccessible.

· Probability theory provides an essential framework for mathematically interpreting and predicting the behavior of complex networks. These include both human designs such as the Internet, power networks, wireless communication, and modern manufacturing systems, as well as natural geophysical systems such as seismic, climatic and hydrologic systems.

· Many current research challenges across the sciences involve a combination of modeling randomness, Monte Carlo simulation and statistical data analysis, all of which depend on probabilistic tools. In particular Statistics and Probability are and have always been inextricably linked.

· The probability theory tries to put the different conjectures about the happening or not happening or not happening of an event into formal quantitative measures.

· The results of probability are very near the actual happening( OR not happening) If experiments are repeated many times and a long time average is computed

· Useful in theoretical distribution: With the help all types of frequency distribution can be prepared. They are prepared on theoretical basis.

· Basis of the theory of sampling: Sometimes the study of a big group is not possible and the cost involved in the study is not just if able. In such cases on the basis of the theory of probability a part of the whole is studied and estimates are made.