# Blog Probability Theory Sampada Vardhe09-07-2021 Data ScienceAdd a Comment Probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty. Most probable event occurs most frequently than events with low probability. If the number of repetitions are small, the frequency of event is to a considerable extent a random quality. We conduct an experiment with great number of repetitions and the frequency of the event becomes less and less random and it stabilizes. If the number of independent trials are sufficiently large, we say that frequency has approached the probability of an event. If an experiment is repeated under essentially homogeneous and similar conditions we generally come across two types of situation  The outcome is unique or certain. The result is not unique but may be one of the several possible outcomes. Random Experiment: An experiment that can result in different outcomes, even though it is repeated in the same manner every time, is called a random experiment. An experiment (or a “trial”) with results that cannot be predicted beforehand, i.e. random. Sample Space: The set of all possible outcomes of a random experiment is called the sample space of the experiment. The sample space is donated by S. Each element of Sample space is called a Sample point. A Sample space can be: Discrete     e.g. S = {1,2,3,4,5,6}             S = {low, medium, high}             S={y, n} Continuous      e.g.  S = R+={x|x>0}              S={x|103 P(A or B) = 1/2 + 1/2 – (1/2 * 2/3) = 2/3 4  Probability Multiplicative Rule: The probability of the combination of two events (sequentially or simultaneously) is equal to the probability of one of them multiplied by the probability of the other provided that the first one has occurred P(A and B) = P(A). P(B/A) where P(B/A) is called conditional probability of event B calculated for the condition that event A has occurred For independent events, P(B/A) = P(B) Two events A and B are called independent if the fact that one event has occurred does not affect the probability that the other event will occur. In such cases: P(A and B) = P(A). P(B) The idea of disjoint events is about whether or not it is possible for the events to occur at the same time. The idea of independent events is about whether or not the events affect each other in the sense that the occurrence of one event affects the probability of the occurrence of the other. If the events are disjoint, then they cannot be independent. A and B disjoint implies that if event A occurs then B does not and vice versa. Knowing that event A has occurred dramatically changes the likelihood that event B occurs – that likelihood is 0. This implies that A and B are not independent. 