(1 votes, average: 1.00 out of 5)What is Probability?
Probability is a branch of mathematics that deals with uncertainty. The term “probability” is used to quantify the degree of belief or confidence that something is true (or false). It gives us the likelihood of occurrence of a given event. It is expressed as a number that could take any value in the closed interval [0,1]
Consider the following experiment describing a simple communication system. A user transmits data through a noisy medium and another user receives it. Here, the sender utters a single alphabet on the phone. Due to the noise characteristics of the communication medium, we do not know whether the user at the destination will be able to hear what the sender has already spoken. Before performing the experiment, we would like to know the likelihood that the user at the destination hears the particular syllable (given the noise characteristics). This likelihood of the particular event is called probability of the event.
Experiment:
Any activity that can produce observable results is called an experiment. For example, tossing a coin (observable results: Head/Tail), Rolling a die (observable results: numbers on the faces of the die), drawing a card from a deck (observable results: symbols, numbers and alphabets on the cards), sending & receiving bits in a communication system (observable results: bits/alphabets transferred or voltage level at the receiver).
Sample Space:
Given an experiment, the sample space comprises a set of all possible outcomes of the experiment. It plays the role of the universal set when modeling the experiment. It is denoted by the letter – ‘S’. Following examples illustrate the sample spaces for various experiments.
Event:
It is also a set of outcomes of an experiment. It is a subset of sample space. Each time the experiment is run, either a particular event occurs or it does not occur. Events are associated with a probability number.
Types of Events:
Events can be classified according to their relationship with one another. The following table shows the classification of events and their definition.
Computing Probability:
The proability of the occurrence of an event (say ‘A’) is given by the ratio of number of ways that particular event can happen and the total number of all possible outcomes.
For example, consider the experiment of an unbiased rolling of a die. The sample space is given by S={1,2,3,4,5,6}. Let’s say that an event is defined as getting ‘4’ when you roll the die. The probability of getting the face with ‘4’ (event) can be calculated as follows.
Axioms of Probability:
Following definitions are assumed for the axioms listed below: ‘S’ denotes the sample space of an experiment, ‘A’ and ‘B’ are events and P(A) denotes the probability of occurrence of event ‘A’.
Properties of Probability:
The definition of probability – has some properties as listed below.
Here the symbol Ø indicates null event, Ā indicates that the event A is NOT occuring.
Joint probability and Marginal probability:
Joint probability is defined as the probability that two or more events occur simultaneously. For two events A and B, the joint probability is denoted by P(A,B) or P(A∩B).
Given two or more events, the marginal probability is the probability of occurrence of a single event. It is also called a-priori probability.
The following table illustrates the concept of computing the joint and marginal probabilities. Here, four events (P, Q, R, S) are used for illustration. For example, the table indicates that the probability of occurrence of both events R & Q is given by b/n. This is the joint probability of R and Q. Adding all the probabilities either row wise or column wise gives us the marginal probability of a single event. For example, adding a/n and b/n gives the marginal probability of event similarly, adding a/n and c/n gives the marginal probability of event P.
Conditional probability or Posteriori probability:
Conditional probabilities (also called posteriori probability) deal with dependent events. It is used to calculate the probability of an event given that some other event has already occurred.
It is denoted as P(B|A)–meaning that ‘the probability of event B given that the event A has occurred already’. It is called “a-posteriori” because it is only available “after” observing A (the first event).
The conditional probability P(B|A) is mathematically computed as
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