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Probability Class 12 :

CBSE Class 12 Maths Notes Chapter 13 Class 12 Notes Maths

CBSE Class 12 Maths Notes Chapter 13 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 A¯.
i.e. A’ = {n : n ∈ S, n ∉ A]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.

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Probability Distribution

If a random variable X takes values X1, X2,…., Xn with respective probabilities P1, P2,…., Pn then

CBSE Class 12 Maths Notes : Probability
is known as the probability distribution of X.

Probability distribution gives the values of the random variable along with the corresponding probabilities.

Mathematical Expectation/Mean

If X is a discrete random variable which assume values X1, X2,…., Xn with respective probabilities P1, P2,…., Pn then the mean x of X is defined as

E(X) = X = P1X1 + P2X2 + … + PnXn = Σni = 1 PiXi

Important Results

(i) Variance V(X) = σ2x = E(X2) – (E(X))2

where, E(X2) = Σni = 1 x2iP(xi)

(ii) Standard Deviation

√V(X) = σx = √E(X2) – (E(X))2