 # Probability

## NCERT Solutions Class 12th Maths by Extramarks covers the whole concept comprehensively. If a student is well versed with NCERT Solutions Class 12th, he/she carries high chances of scoring more. As the CBSE Class 12 syllabus is gigantic, it is important to be prepared for the upcoming challenges.  Class 12 Maths in the CBSE curriculum consists of a very interesting chapter called ‘Probability’. The chapter is explained through Animations, Mindmaps, Virtual Lab, etc. to help student understand the topic at its best.  Extramarks, the best medium of online education in India & abroad serves all learning purpose for Classes K-12 and provides the best study solutions. A random variable is a real valued function whose domain is the sample space of a random experiment. Trials of a random experiment are called Bernoulli trials if they satisfy the following conditions: (i) There should be a finite number of trials. (ii) The trials should be independent. (iii) Each trial has exactly two outcomes, i.e., success or failure. (iv) The probability of success remains the same in each trial. P(x) = ncxqn – xpx, where x = 0, 1, 2, ... n, is called the probability function of binomial distribution. A binomial distribution with n-Bernoulli trials and probability of success in each trial as p is denoted by B(n, p). The conditional probability of the event E given that F has already occurred, is given by P(E | F) = P(E  F)/P(F), where P(F) is not equal to 0. 0 ≤ P(E | F) ≤ 1 P(Eʹ | F) = 1 – P(E | F) P[(A  B) | F] = P(A | B) + P(B | F) – P[(A  B) | F] P(E  F) = P(F) P(E | F) = P(E) P(F | E), where P(E) and P(F) are not equal to 0. If E and F are independent events P(E  F) = P(E).P(F). Keywords: Probability Distribution of Random Variable, Expectation of Random Variable, Variance of X of Random Variable, Standard Deviation of Random Variable, Theorem of Total Probability, Bayes’ Theorem

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