# Probability Class 11 Notes Mathematics Chapter 16 - CBSE

## Chapter : 16

## What Are Probability ?

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## Probability

Probability is a concept which numerically measures the degree of uncertainty.

## Events

A subest of sample space associated with a random experiment is called an event generally denoted by ‘E’.

**For example:** When a die as thrown once the sample space, S = {1, 2, 3, 4, 5, 6}

Then E = {2, 4, 6, } i.e., event of getting an even number

Clearly E ⊆ S

## Occurrence Of An Event

An event associated any one of the elementary events associated to it is an outcome of the the experiment.

**For example:** Suppose a die is thrown and A be an event of getting an odd number.

Then A = {1, 3, 5}

## Types Of Events

**Simple Event:**An event which has only one sample point of the sample space, it is called a simple event or elementary event.**Compound Event:**An event which is not simple is called a compound event, composite event or mixed event.

**For example:** In simultaneous toss of two coins we have sample space as {HH, HT, TH, TT}. Then event of getting a tail on both the coins is a simple event. And event of getting atleast one tail is a compound event.

**Sure Event:**The event which is certain to happen, is called a sure event.

**For example:** On throwing a die, the sample space is {1, 2, 3, 4, 5, 6} Then, event of getting a number less than 7 is a sure event.

**Impossible Event:**The event which has no element on empty is called an impossible event or null event. So, f Ì S, as f is an event.

**For example:** On throwing a die once, we have Sample space as {1, 2, 3, 4, 5, 6}.

Then, E = event of getting a number less than 1, is an impossible event, since E = Φ.

**Mutually Exclusive Event:**Two event are said to be mutually exclusive events if the occurrence of any one of the excludes the occurrence of the other event i.e., they cannot occur simultaneously.

Let E_{1}and E_{2}be the events, if E_{1}∩ E_{2}= f, then E_{1}and E_{2}are called the mutually exclusive events.

**For example:** When we throw a die, we have

S = {1, 2, 3, 4, 5, 6}

Let E_{1} = event of getting a prime number

E_{2} = event of getting a composite number

Then, E_{1} = {2, 3, 5}

and E_{2} = { 4, 6}

Clearly, E_{1} ∩ E_{2} = Φ

Hence E_{1} and E_{2} are mutually exclusive.

**Independent Event:**Two events are said to be independent if the occurrence of one does not depend upon the occurrence of other. If two events are not independent, then they are dependent.

**For example:** When we toss two unbaised coine

Then E_{1} = Event of getting a tail on the first coin

E_{2} = Event of getting a tail on the second coin

Clearly, the occurrence of a tail on the second coin does not depend upon the occurrence of a tail on the first coin.

∴ E_{1} and E_{2} are independent event.

**Equally Likely Event:**Events are said to be equally likely if none of the events is expected to occur in preference to the others.

**For example:** On rolling an unbiased die, each outcome is equally likely to happen.

**Complementary Events:**In random experiment let S be a sample space and let E be an event.

Then E ⊆ S

Now, E ⊆ S \Rightarrow E^{C} ⊆ S

Then E^{C} or E' is also an event, called the complement of E. The complement event of E the E' or E^{C} consist of those outcomes that do not correspond to the occurrence of E·E' is also called the event 'not E' when two unbaised coins are thrown.

S = {HH,HT, TH, TT}

Let E = {HT, TH} = event of getting only one head

E' = {HH, TT}

## Algebra Of Events

Consider a random experiment of sample space S.

Let E ⊆ S and F ⊆ S

Then, E as well as F is an event.

- (E ∩ F) is an event that occurs only when each of E and F occurs.
- (E ∪ F) is an event that occurs only when at least one of E and F occurs.
- $$\bar{\text{E}}\space\text{is an event that occurs}$$ only when E does not occur.

## Axiomatic Approach To Probability

**Probability of an event:**In a random experiment, let S be a sample space and E ⊆ S. Then, E is an event.

The probability of occurrence of E is defined as :

$$\text{P(E)} =\\\frac{\text{Number of outcomes favourable to occurrence of E}}{\text{Number of all possible outcomes}}\\\text{P(E)}=\\\frac{\text{No. of distinct elements in E}}{\text{No. of distinct elements in S}}\\=\frac{\text{n(E)}}{\text{n(S)}}$$

**Probability of Equally Likely Outcome:**All the outcomes are said to be equal if the chances of occurrence of each simple event is same. Consider an experiment, whose sample space be S.

i.e., S = {e_{1} , e_{2} , .... e_{n} }

$$\text{Since}\space \displaystyle\sum^n_{i=1}\text{P}(e_{i}) = 1\\\therefore\space \text{P +P+P+....+P}(\text{n times}) = 1\\\text{Or\space}\text{nP = 1}\\\text{P =}\frac{1}{n}$$

**Odds of an event:**Let there be m outcomes favourable to an event and n outcomes unfavourable to E. Then,

(i) Odds in favour of

$$\text{E} =\frac{m}{n}\text{or (m : n)}\\\text{(ii) Odds against}\\\text{E} =\frac{n}{m}\text{or (n : m)}\\\text{(iii)}\space \text{P(E)}=\frac{m}{\text{m+n}}$$

**Probability of Occurrence of Complement of an Event:**$$\text{If}\space \bar{E}\space\text{be an event an}\space \bar{\text{E}}\\\text{be its complement, then}\\\text{P(}\bar{\text{E}}) + \text{P(E) = 1.}$$**Probability of the event ‘A’ or ‘B’:**(i.e., Addition rule of probability)- If E and F are two events associated with a random experiment, then

P (E or F) = P (E) + P (F) – P (E and F)

But when two events E and F are mutually

exclusive then

P (E or F) = P (E) + P (F)

and when events E and F are mutually exclusive and exhaustive, than

P (E or F) = P (E) + P (F) = 1

## Important Points

- A sample is a discrete sample space, if S has a finite set.
- The number of event of a sample sapce S is 2
^{n}, where ‘n’ is the number of elements in S. - Simple events of a sample space are mutually exclusive.
- E and E' (E
^{C}) are mutually exclusive and exhaustive events.

i.e., E ∩ E' = Φ

and E ∪ E' = f

or P (E ∩ E') = P (S)

P (E) + P (E') = 1

P (E') = P (not E) = 1 – P (E)