Probability
In day to day life, everyone uses the word probability. Generally, however, most people do not have a definite idea about the meaning of probability. The origin of the probability theory starts from the study of games like cards, tossing coins, dice, etc. But in modern times, probability has great importance in decision making. According to the classical theory, probability is the ratio of the favorable case to the total number of equally likely cases. Empirical or relative frequency probability is based on logic, past experience and present condition. According to the subjective approach, the probability of an event is assigned by an individual on the basis of evidences available to him/her.
Some basic concepts in Probability:
Experiment: When we conduct a trial or experiment to obtain some statistical information, it is called an experiment.
Event: In an experiment, the outcome is called the event.
Exhaustive event: In a probability experiment, the total outcome is called the exhaustive event.
Equally –likely event: The events are said to be equally –likely if the chance of happening is equal of all events. In other words, events are said to be equally likely when one event does not occur more often than the others.
Mutually exclusive events: In the case of probability, two events are said to be mutually exclusive when they cannot occur simultaneously in a single trial.
Complementary event: In probability, when event A and event B are mutually exclusive and exhaustive, then Event A is called the complementary event of B, and Event B is called the complementary event of A.
Simple and Compound event: When we consider the probability of occurrence or no occurrence of a single event, then it is said to be a simple event. In the case of a compound event, we consider the probability of the joint occurrence of two or more events.
Dependent event: When the probability of an occurrence of one event affects the probability of the occurrence of another event, then it said to be the dependent event.
Independent event: When the probability of an occurrence of one event does not affect the probability of an occurrence of another event, then it is said to be the independent event.
Theorems of probability:
In the case of probability, the following theorems are applied:
1. Addition theorem: Addition theorem states that if A and B are two mutually exclusive events, then the probability of occurrence of either A or B is the sum of the individual probability of A and B. In mathematics, we can denote the addition theorem as:
P (A or B) =P (A) +P (B).
2. Multiplication theorem in case of independent event: According to the multiplication theorem, if A and B are two independent events, then the probability of the simultaneous occurrence of A and B is equal to the product of their individual probability. In mathematics, we can denote the multiplication theorem as:
P (AB) =P (A)*P (B).
Multiplication theorem in case of dependent event: The multiplication rule of probability states that if event A and B are two dependent events, then the probability of their occurrence is equal to the probability of one event multiplied by the conditional probability of the second event. Mathematically we can denote this multiplication theorem as:
P (AB) = P (A)*P (B/A) or
P (AB) =P (B)*P (A/B)
3. Bayes’ theorem of probability: Bayes’ theorem of probability states that in an experiment, if we know the probability of an occurrence of an event, then the probability of its occurrence exactly 1,2,3,….r times in n trials can be determined by using this formula:
Where r=1, 2, 3…….n
P(r) = probability of r successes in n trials.
P = probability of success or happening of an event in one trail.
q = probability of failure or not happening of the event in one trail.
n = total number of trials.
