Class 12 Maths Revision Notes for Probability of Chapter 13

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  B.
• The event A and B: The set A  B.
• The event A but not B: A – B.
• Mutually exclusive events: A and B are mutually exclusive if A  B = .
• Exhaustive and Mutually exclusive events: Events E₁, E₂,…….., E_n are mutually exclusive and exhaustive if E₁ E₂ ……. E_n = S and E_i  E_j =  for all .
• 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)  0
(b) P(S) = 1
(c) If E and F are mutually exclusive event, then P(E  F) = P(E) + P(F)
If E₁, E₂, E₃………… are n mutually exclusive events, then 

• Probability of an event in terms of the probabilities of the same points (outcomes): Let S be the sample space containing n exhaustive outcomes  i.e., S = 

Now from the axiomatic definition of the probability:
(a) 0  P(W_i) 1, for each .
(b) P(W₁) + P(W₂) + …….+ P(W₃) = P(S) = 1
(c) For any event A, P(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) = 
where  = Number of elements in the set A. and  = Number of elements in set S.

• If A is any event, then P(not A) = 1 – P(A)    
• 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