Class 12 Class 12 Maths Class 12 Maths Notes

CBSE Class 12 Maths Notes Probability

Probability
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Probability is part of Class 12 Maths Notes for Quick Revision. Here we have given Class 12 Maths Notes Probability.

Probability

Probability

Event: A subset of the sample space associated with a random experiment is called an event or a case.
e.g. In tossing a coin, getting either head or tail is an event.

Equally Likely Events: The given events are said to be equally likely if none of them is expected to occur in preference to the other.
e.g. In throwing an unbiased die, all the six faces are equally likely to come.

Mutually Exclusive Events: A set of events is said to be mutually exclusive, if the happening of one excludes the happening of the other, i.e. if A and B are mutually exclusive, then (A ∩ B) = Φ
e.g. In throwing a die, all the 6 faces numbered 1 to 6 are mutually exclusive, since if any one of these faces comes, then the possibility of others in the same trial is ruled out.

Exhaustive Events: A set of events is said to be exhaustive if the performance of the experiment always results in the occurrence of at least one of them.
If E1, E2, …, En are exhaustive events, then E1 ∪ E2 ∪……∪ En = S.
e.g. In throwing of two dice, the exhaustive number of cases is 62 = 36. Since any of the numbers 1 to 6 on the first die can be associated with any of the 6 numbers on the other die.

Complement of an Event: Let A be an event in a sample space S, then the complement of A is the set of all sample points of the space other than the sample point in A and it is denoted by A’or \bar { A }

.
i.e. A’ = {n : n ∈ S, n ∉ A]

Note:
(i) An operation which results in some well-defined outcomes is called an experiment.
(ii) An experiment in which the outcomes may not be the same even if the experiment is performed in an identical condition is called a random experiment.

Probability of an Event

If a trial result is n exhaustive, mutually exclusive and equally likely cases and m of them are favourable to the happening of an event A, then the probability of happening of A is given by
Probability Class 12 Notes Maths Chapter 13 1

Note:
(i) 0 ≤ P(A) ≤ 1
(ii) Probability of an impossible event is zero.
(iii) Probability of certain event (possible event) is 1.
(iv) P(A ∪ A’) = P(S)
(v) P(A ∩ A’) = P(Φ)
(vi) P(A’)’ = P(A)
(vii) P(A ∪ B) = P(A) + P(B) – P(A ∩ S)

Conditional Probability: Let E and F be two events associated with the same sample space of a random experiment. Then, probability of occurrence of event E, when the event F has already occurred, is called a conditional probability of event E over F and is denoted by P(E/F).
Probability Class 12 Notes Maths Chapter 13 2
Similarly, conditional probability of event F over E is given as
Probability Class 12 Notes Maths Chapter 13 3

Properties of Conditional Probability: If E and E are two events of sample space S and G is an event of S which has already occurred such that P(G) ≠ 0, then
(i) P[(E ∪ F)/G] = P(F/G) + P(F/G) – P[(F ∩ F)/G], P(G) ≠ 0
(ii) P[(E ∪ F)/G] = P(F/G) + P(F/G), if E and F are disjoint events.
(iii) P(F’/G) = 1 – P(F/G)
(iv) P(S/E) = P(E/E) = 1

Multiplication Theorem: If E and F are two events associated with a sample space S, then the probability of simultaneous occurrence of the events E and F is
P(E ∩ F) = P(E) . P(F/E), where P(F) ≠ 0
or
P(E ∩ F) = P(F) . P(F/F), where P(F) ≠ 0
This result is known as multiplication rule of probability.

Multiplication Theorem for More than Two Events: If F, F and G are three events of sample space, then
Probability Class 12 Notes Maths Chapter 13 4

Independent Events: Two events E and F are said to be independent, if probability of occurrence or non-occurrence of one of the events is not affected by that of the other. For any two independent events E and F, we have the relation
(i) P(E ∩ F) = P(F) . P(F)
(ii) P(F/F) = P(F), P(F) ≠ 0
(iii) P(F/F) = P(F), P(F) ≠ 0
Also, their complements are independent events,
i.e. P(\bar { E }\bar { F }) = P(\bar { E }) . P(\bar { F })
Note: If E and F are dependent events, then P(E ∩ F) ≠ P(F) . P(F).

Three events E, F and G are said to be mutually independent, if
(i) P(E ∩ F) = P(E) . P(F)
(ii) P(F ∩ G) = P(F) . P(G)
(iii) P(E ∩ G) = P(E) . P(G)
(iv)P(E ∩ F ∩ G) = P(E) . P(F) . P(G)
If atleast one of the above is not true for three given events, then we say that the events are not independent.
Note: Independent and mutually exclusive events do not have the same meaning.

Baye’s Theorem and Probability Distributions

Partition of Sample Space: A set of events E1, E2,…,En is said to represent a partition of the sample space S, if it satisfies the following conditions:
(i) Ei ∩ Ej = Φ; i ≠ j; i, j = 1, 2, …….. n
(ii) E1 ∪ E2 ∪ …… ∪ En = S
(iii) P(Ei) > 0, ∀ i = 1, 2,…, n

Theorem of Total Probability: Let events E1, E2, …, En form a partition of the sample space S of an experiment.If A is any event associated with sample space S, then
Probability Class 12 Notes Maths Chapter 13 5

Baye’s Theorem: If E1, E2,…,En are n non-empty events which constitute a partition of sample space S, i.e. E1, E2,…, En are pairwise disjoint E1 ∪ E2 ∪ ……. ∪ En = S and P(Ei) > 0, for all i = 1, 2, ….. n Also, let A be any non-zero event, the probability
Probability Class 12 Notes Maths Chapter 13 6

Random Variable: A random variable is a real-valued function, whose domain is the sample space of a random experiment. Generally, it is denoted by capital letter X.
Note: More than one random variables can be defined in the same sample space.

Probability Distributions: The system in which the values of a random variable are given along with their corresponding probabilities is called probability distribution.
Let X be a random variable which can take n values x1, x2,…, xn.
Let p1, p2,…, pn be the respective probabilities.
Then, a probability distribution table is given as follows:
Probability Class 12 Notes Maths Chapter 13 7
such that P1 + p2 + P3 +… + pn = 1
Note: If xi is one of the possible values of a random variable X, then statement X = xi is true only at some point(s) of the sample space. Hence ,the probability that X takes value x, is always non-zero, i.e. P(X = xi) ≠ 0

Mean and Variance of a Probability Distribution: Mean of a probability distribution is
Probability Class 12 Notes Maths Chapter 13 8

Bernoulli Trial: 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.

Binomial Distribution: The probability distribution of numbers of successes in an experiment consisting of n Bernoulli trials obtained by the binomial expansion (p + q)n, is called binomial distribution.
Let X be a random variable which can take n values x1, x2,…, xn. Then, by binomial distribution, we have P(X = r) = nCr prqn-r
where,
n = Total number of trials in an experiment
p = Probability of success in one trial
q = Probability of failure in one trial
r = Number of success trial in an experiment
Also, p + q = 1
Binomial distribution of the number of successes X can be represented as
Probability Class 12 Notes Maths Chapter 13 9

Mean and Variance of Binomial Distribution

(i) Mean(μ) = Σ xipi = np
(ii) Variance(σ2) = Σ xi2 pi – μ2 = npq
(iii) Standard deviation (σ) = √Variance = √npq
Note: Mean > Variance

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Class 12 Maths Notes

Relations and Functions

Inverse Trigonometric Functions

Matrices

Determinants

Continuity and Differentiability

Application of Derivatives

Integrals

Application of Integrals

Differential Equations

Vector Algebra

Three Dimensional Geometry

Linear Programming

Probability

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