Probability Class 12 Notes Mathematics

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Class 12 Mathematics notes Chapter 13 Probability

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CBSE Class 12 Mathematics Chapter 13 Probability

Sample Space: The set of all possible outcomes of a random experiment. It is denoted by the symbol S.
Sample points: Elements of the sample space.
Event: A subset of the sample space.
Impossible Event: The empty set.
Sure Event: The whole sample space.
Complementary event or “not event”: The set “S” or S – A.
The event A or B: The set A $\cup$ B.
The event A and B: The set A $\cap$ B.
The event A but not B: A – B.
Mutually exclusive events: A and B are mutually exclusive if A $\cap$ B = $\phi$ .
Exhaustive and Mutually exclusive events: Events E₁, E₂,…….., E_n are mutually exclusive and exhaustive if E₁ $\cup$ E₂ $\cup$ ……. $\cup$ E_n = S and E_i $\cap$ E_j = $\phi$ for all $i \ne j$ .
Exiomatic approach to probability: To assign probabilities to various events, some axioms or rules have been described.

Let S be the sample space of a random experiment. The probability P is a real values function whose domain is the power set of S and range is the interval [0, 1] satisfying the following axioms:

(a) For any event E, P(E) $\ge$ 0

(b) P(S) = 1

If E₁, E₂, E₃………… are n mutually exclusive events, then ${\rm{P}}\left( {\mathop \cup \limits_{i = 1}^n {{\rm{E}}_i}} \right) = \sum\limits_{i = 1}^n {{\rm{P}}\left( {{{\rm{E}}_i}} \right)}$

Probability of an event in terms of the probabilities of the same points (outcomes): Let S be the sample space containing n exhaustive outcomes ${{\rm{W}}_1},{{\rm{W}}_2},{{\rm{W}}_3},......{{\rm{W}}_n}$ i.e., S = $\left( {{{\rm{W}}_1},{{\rm{W}}_2},{{\rm{W}}_3},......{{\rm{W}}_n}} \right)$

Now from the axiomatic definition of the probability:

(a) 0 $\le$ P(W_i) $\le$ 1, for each ${{\rm{W}}_i} \in {\rm{S}}$ .

(b) P(W₁) + P(W₂) + …….+ P(W₃) = P(S) = 1

Equally likely outcomes: All outcomes with equal probability.
Classical definition of the probability of an event: For a finite sample space with equally likely outcome, probability of an event A.

P(A) = ${{n\left( {\rm{A}} \right)} \over {n\left( {\rm{S}} \right)}}$

where $n\left( {\rm{A}} \right)$ = Number of elements in the set A. and $n\left( {\rm{S}} \right)$ = Number of elements in set S.

If A is any event, then P(not A) = 1 – P(A) $\Rightarrow$ ${\rm{P}}\left( {\overline {\rm{A}} } \right) = 1 - {\rm{P}}\left( {\rm{A}} \right)$ $\Rightarrow$ ${\rm{P}}\left( {{\rm{A'}}} \right) = 1 - {\rm{P}}\left( {\rm{A}} \right)$
The conditional probability of an event E, given the occurrence of the event F is given by
·
·
Theorem of total probability:

be a partition of a sample space and suppose that each of has non zero probability. Let A be any event associated with S, then

Bayes’ theorem: If are events which constitute a partition of sample space S, i.e. are pairwise disjoint and be any event with non-zero probability, then,
Random variable: A random variable is a real valued function whose domain is the sample space of a random experiment.
Probability distribution: The probability distribution of a random variable X is the system of numbers

Where,

Mean of a probability distribution: Let X be a random variable whose possible values occur with probabilities respectively. The mean of X, denoted by is the number . The mean of a random variable X is also called the expectation of X, denoted by E (X).
Variance: Let X be a random variable whose possible values occur with probabilities respectively. Let be the mean of X. The variance of X, denoted by Var (X) or is defined as or equivalently . The non-negative number, is called the standard deviation of the random variable X.

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 remains the same in each trial.

For Binomial distribution

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CBSE Class-12 Revision Notes and Key Points

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