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 E₁ and E₂ be the events, if E₁ ∩ E₂ = f, then E₁ and E₂ are called the mutually exclusive events.

For example: When we throw a die, we have

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

Let E₁ = event of getting a prime number

E₂ = event of getting a composite number

Then, E₁ = {2, 3, 5}

and E₂ = { 4, 6}

Clearly, E₁ ∩ E₂ = Φ

Hence E₁ and E₂ 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 E₁ = Event of getting a tail on the first coin
E₂ = 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.

∴ E₁ and E₂ 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 = {e₁ , e₂ , .... 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 2ⁿ , 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)