Probability ML Aggarwal ISC Class-12 Maths Solutions Chapter-10. Step by step Solutions of ML Aggarwal ISC Class 12 Mathematics for Exercise Questions with Chapter Test. Visit official Website CISCE for detail information about ISC Board Class-12 Mathematics.

## Probability ML Aggarwal ISC Class-12 Maths Solutions Chapter-10

Board | ISC |

Class | 12 |

Subject | Mathematics |

Chapter-10 | Probability |

Session | 2024-25 |

Topics | Solutions of ML Aggarwal |

### 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 E_{1}, E_{2}, …, E_{n} are exhaustive events, then E_{1} ∪ E_{2} ∪……∪ E_{n} = S.

e.g. In throwing of two dice, the exhaustive number of cases is 6^{2} = 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]

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.

**Conditional Probability:**

Let E and F be two events associated with the same sample space of a random experiment.

**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.

**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.

**Exercise – 10.1**

Probability ML Aggarwal ISC Class-12 Maths Solutions

### Exercise – 10.2

Probability ISC Class-12 Maths Solutions Chapter-10

### Exercise – 10.3

Probability ISC Class-12 Maths Solutions Chapter-10

### Exercise – 10.4

Probability ML Aggarwal ISC Class-12 Maths Solutions Chapter-10

### Exercise – 10.5

Probability Class-12 Maths Solutions Chapter-10

### Exercise – 10.6

Probability Class-12 Maths Solutions Chapter-10

### Exercise – 10.7

Probability Class-12 Maths Solutions Chapter-10

### Exercise – 10.8

Ch-10 Probability ISC Class-12 Maths Solutions

### Exercise – 10.9

Probability Class-12 Maths Solutions Chapter-10

### Exercise – 10.10

Probability ISC Class-12 Maths Solutions Chapter-10

### Chapter Test

Probability ML Aggarwal ISC Class-12 Maths Solutions Chapter-10

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