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NCERT Class 12- Mathematics: Chapter β 13 Probability Part 1
13.1 Overview
13.1.1 Conditional Probability
If E and F are two events associated with the same sample space of a random experiment, then the conditional probability of the event E under the condition that the event F has occurred, written as , is given by
13.1.2 Properties of Conditional Probability
Let E and F be events associated with the sample space S of an experiment. Then:
(i)
(ii) , where A and B are any two events associated with S.
(iii)
13.1.3 Multiplication Theorem on Probability
Let E and F be two events associated with a sample space of an experiment. Then
If and G are three events associated with a sample space, then
13.1.4 Independent Events
Let E and F be two events associated with a sample space S. If the probability of occurrence of one of them is not affected by the occurrence of the other, then we say that the two events are independent. Thus, two events E and F will be independent, if
(a)
(b)
Using the multiplication theorem on probability, we have
(c)
Three events A, B and C are said to be mutually independent if all the following conditions hold:
and
13.1.5 Partition of a Sample Space
A set of events is said to represent a partition of a sample space S if
(a)
(b) , and
(c) Each
13.1.6 Theorem of Total Probability
Let be a partition of the sample space S. Let A be any event associated with S, then
13.1.7 BayesΥ Theorem
If are mutually exclusive and exhaustive events associated with a sample space, and A is any event of non-zero probability, then
13.1.8 Random Variable and Its Probability Distribution
A random variable is a real valued function whose domain is the sample space of a random experiment.
The probability distribution of a random variable X is the system of numbers
Where
13.1.9 Mean and Variance of a Random Variable
Let be a random variable assuming values with probabilities respectively such that . Mean of , denoted by [expected value of X denoted by ] is defined as
and variance, denoted by , is defined as
Standard deviation of the random variable is defined as
13.1.10 Bernoulli Trials
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: success or failure
(iv) The probability of success (or failure) remains the same in each trial.
13.1.11 Binomial Distribution
A random variable taking values is said to have a binomial distribution with parameters and , if its probability distribution is given by