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CBSE Guide Probability class 12 Notes Mathematics
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Class 12 Mathematics notes Chapter 13 Probability
Download CBSE class 12th revision notes for chapter 13 Probability in PDF format for free. Download revision notes for Probability class 12 Notes and score high in exams. These are the Probability class 12 Notes prepared by team of expert teachers. The revision notes help you revise the whole chapter 13 in minutes. Revision notes in exam days is one of the best tips recommended by teachers during exam days.
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,…….., En are mutually exclusive and exhaustive if E1 E2 ……. En = S and Ei Ej = 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(Wi) 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.
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
- 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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