# Probability Class 12 Notes Mathematics

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## CBSE Guide 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,…….., En are mutually exclusive and exhaustive if E1$\cup$ E2 $\cup$…….$\cup$ En = S and Ei $\cap$ Ej = $\phi$ 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

(c) If E and F are mutually exclusive event, then P(E $\cup$ F) = P(E) + P(F)

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(Wi) $\le$1, for each ${{\rm{W}}_i} \in {\rm{S}}$.

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

(c) For any event A, P(A) = $\sum {{\rm{P}}\left( {{{\rm{W}}_i}} \right),}$ ${{\rm{W}}_i} \in {\rm{A}}$

• 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