**Probability** is one of the most important topics in your preparation for JEE Main 2019 as it certain that this topic fetches you **2-3 questions**. It requires a good amount of practice of standard questions asked in previous year JEE Main, JEE Advanced and BITSAT. Download the Probability notes pdf from the link given at the end of the article.

**1. Random Experiment**

An experiment where all possible outcomes or results are well known in advance but exactly what is going to occur that cannot be predicted before completion of the experiment is called a random experiment.

**2. Sample Space**

The set of all possible outcomes or results of a random experiment is said to be the sample space of the random experiment. It is denoted by S.

For example, in the random experiment of tossing an unbiased coin the sample space is the **set S = {H, T}**

In the random experiment of tossing two unbiased coins the sample space is

**S = {HH, HT, TH, TT}**

**3.** **Event**

A subset of the sample space is called as

### (a) Mutually Exclusive Events

Two or more events are said to be mutually exclusive if the occurrence of any one of them implies the non-occurrence of the other i.e. they cannot occur simultaneously

Events A and B will be mutually exclusive if and only if **P(A∩B) = 0**

### (b) Exhaustive Events

Several events are said to be exhaustive if they include all possible outcomes of the sample space of the random experiment i.e., at least one of them must occur if the experiment is performed

If A_{1}, A_{2}, A_{3},...A_{n} are exhaustive events, then A_{1} U A_{2} U A_{3} ….U A_{n} where S is the sample space

**(c) Equally Likely Events**

Two or more events are said to be equally likely if chances of occurrence of any one of them have no preference in comparison to chances of occurrence of any one of the rests.

**4. Classical Definition of Probability**

In a random experiment if n number of mutually exclusive, exhaustive and equally likely outcomes or results are there out of which m number of outcomes or results are favourable to an event A then the probability of occurrence of event A is defined as P(A)=m/n

For any events 0

If P(A)=0, A is a null or impossible event

If P(A)=1, A is a sure event

**5. Complimentary Event**

The complimentary event of the event A is denoted by A’. If P(A) represents the probability of occurrence of the event A, then non-occurrence of the event A is represented by P(A’).

Complementary events are always mutually exclusive and exhaustive

P(A)+P(A’) = 1

**6. Use of set theory in probability**

**(a)**Union of events

**(b)** Intersection of events

**(c) **For any 2 events A and B, P(AUB) = P(A) + P(B) - P(A B)

**(d) **For any 2 events A and B, P(AUB) ≤ P(A)+P(B)

**(e) **If A and B are mutually exclusive events, then P(A∩B) = 0 and P(AUB)=P(A)+P(B)

**(f)** For any three events A, B and C,

P(A+B+C) = P(A) + P(B) + P(C) - P(AB) - P(BC) - P(CA) + P(ABC)

**(g) **If A_{1}, A_{2}, A_{3},...A_{n }are mutually exclusive events, then P (A_{1 }+ A_{2} + A_{3 }+ …. +A_{n} ) = P (A_{1}) + P (A_{2}) +……..+ P (A_{n})

**(h) **If A_{1}, A_{2}, A_{3},...A_{n }are mutually exclusive and exhaustive events, then P (A_{1}) + P (A_{2}) +……. + P (A_{n}) =

**7. Conditional Probability**

If an event B has already occurred, the probability of occurrence of event A is represented by P(A|B). Thus P(A|B) = Probability of occurrence of event A under the assumption that event B has already occurred. Obviously, P(A|B) is conditional and it may depend on the occurrence or non-occurrence of the event B.

(a) P(A|B) =P(AB) / P(B)

(b) P(B|A) =P(AB) / P(A)

(c) P(AB)=P(A|B). P(B)=P(B|A) . P(A)

### Independent Events

Two events A and B are said to be independent events if occurrence or non-occurrence of A has no impact on occurrence or non-occurrence of B and vice versa.

When events are taken from different random experiments, then they become independent.

If A and B are independent events, then P(A|B) = P(A) and P(B|A) =P(B)

If A and B are independent events, then P(AB)=P(A). P(B)

### Mutually Independent Events

Three events are said to be mutually independent if P(ABC)=P(A)P(B)P(C)

This can be generalized for n events

### Pairwise Independent Events

Three events A, B, C are said to be pairwise independent if

P(AB)=P(A)P(B), P(BC)=P(B)P(C), P(AC)=P(A)P(C)

Mutually independent events are necessarily pairwise independent

Pairwise independent events are NOT necessarily mutually independent

**8. Theorem of total probability**

** **If A_{1}, A_{2}, A_{3},...A_{n} be n mutually exclusive and exhaustive events and B be an event which can occur if and only if any one of A_{1}, A_{2}, A_{3},...A_{n} does occur. Then the probability of occurrence of event B is

**9. Bayes’ Theorem**

** **If A_{1}, A_{2}, A_{3},...A_{n} be n mutually exclusive and exhaustive events and B be an event which can occur if and only if any one of A_{1}, A_{2}, A_{3},....A_{n} does occur. Then the probability of occurrence of an event under the assumption that event B has already occurred is given by

**10. Binomial Distribution**

Any random experiment which two mutually exclusive and exhaustive outcomes has only (called success and failure) when it is repeated a finite number of times, the probability distribution obtained is the binomial distribution.

Let p = probability of success in a single trial

q = probability of failure in a single trial

Such that p + q = 1

And n=number of times the trial has been repeated

Then the probability of getting exactly x number of successes out of these n number of trials is given by

P (X = x) = ^{n}C_{x} p^{x }q^{n - x} where x=0,1,2, 3, …... n

The probabilities P(X=x) are given by the term of the binomial expansion (p + q)^{n }. Because of this, the probability distribution is called the binomial distribution

The function P(x) is called a probability mass function because x can take only discrete values

P (X = 0) + P (X=1) ……. + P (X = n) = 1

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write a commentSurender KaurNov 25, 2018

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