CBSE Class 11 Maths Revision Notes Chapter 16

Class 11 Mathematics Revision Notes for Probability of Chapter 16

Probability is a relatively easy chapter in Class 11 Mathematics. Students can score better if this topic is studied well. Extramarks Revision Notes for Class 11 Mathematics Chapter 16 Probability include all the necessary topics to ensure retention of information. It contains important concepts written in a concise and simple manner. 

Class 11 Mathematics Revision Notes for Probability of Chapter 16 – Free Download

Access Class 11 Mathematics Chapter 16 – Probability Notes

Probability:

Probability is a numerical indicator of the degree of uncertainty in a variety of occurrences. It is a measure of how likely an event is to occur. It is defined mathematically as the ratio of the number of favourable outcomes of an event to the total number of outcomes of the event. The formula for calculating the probability of an event for an experiment with a total ‘n’ number of possible outcomes, the ‘x’ number of favourable outcomes is as follows.

Probability=number of favourable outcomes / total number of outcomes

In an experiment, the probability value of any event will always be between 0 and 1. Where the probability of 0 indicates that the event will not occur, a probability of 1 indicates that the event will occur every time.

In an experiment, the chance of any outcome can never be negative.

The words ‘probably,’ ‘doubt, “most probably,’ ‘chances,’ and so on all include ambiguity.

Approaches to Probability: 

• Statistical approach: It includes observation and collection of data.
• Classical approach: It considers only events with equal probabilities.
• Axiomatic method: For real-life circumstances. It is strongly related to set theory.

Random Experiments:

• There are numerous possibilities.
• It is difficult to predict the outcome in advance.

Outcomes: 

An outcome is the most probable result of a random experiment.

The set of all possible outcomes of a random experiment is referred to as sample space. It is represented by the letter S. In a coin flip, for example, the sample space is Head, Tail.

A sample point is a distinct element of the sample space. In a coin flip, for example, the head is a sample point.

Event:

Any sequence of results from a random experiment constitutes an event. The sample space of an experiment contains all feasible outcomes. The events are thus subsets of the sample space, as is evident.

An event is a subset E of the sample space S. For instance when you roll the dice and the result is unexpected.

Occurrence of an event: 

An event E is said to have occurred in a sample space S if the experiment’s outcome ω is such that ω∈E. If the outcome ω is such that ω∈E, we say that event E did not occur.

Types of Events:

Impossible and Sure Events: The empty set ϕ and the sample space S depict events that are both impossible and certain. The impossible occurrence is represented by ϕ, and the complete sample space is called a Sure Event. 

When rolling dice, for example, an impossible event is when the number is larger than 6, and a sure event is when the number is less than or equal to 6.

Simple (or elementary) event: A simple event just has one sample point in the sample space.

There are exactly n simple occurrences in a sample space with n different objects. For instance, getting a four on a dice throw could be a straightforward occurrence.

Compound Event: An event with many sample points is referred to as a compound event. A compound event, for instance, is the likelihood of rolling an odd number on a die and then tossing a coin to land on tails. 

Algebra of Events:

1. Complementary Event
2. Events ‘A’ and ‘B’
3. Event ‘A but not B’
4. Event ‘A’ or ‘B’

Complementary Event:

Complementary event to A= ′ not A ′   

For example, if event A=Event of getting an odd number in the throw of a die, which is {1, 3, 5}, then Complementary event to A= Event of  getting an even number in a throw of the die, that is {2, 4, 6}   

A ′ ={ ω : ω ∈S and ω∉A}=S−A (S is the Sample Space)  

Event (A or B): 

A∪B is the union of two sets A and B; it contains all of the elements that are present in both sets.

If sets A and B represent two events in a sample space, then A∪B represents the event ‘either A or B or both’. This event A∪B is also known as A or B Event 

A or B = A∪B = { ω:ω ∈A or ω∈B }

Event ‘A and B’:

A∩B is the intersection of two sets A and B; it contains all of the elements that are shared by both sets, ie., which belong to both A and B. If A and B are two events, the set A∩B represents the events A and B.

Thus, A ∩ B = { ω : ω ∈A and ω ∈B }  

Event ‘A but not B’:

A–B is the set of all the items in A that are not in B. As a result, the set A–B may denote the event ‘A but not B’.

A — B = A ∩ B ′   

Mutually exclusive events:

Events A and B are said to be mutually exclusive if the occurrence of one limits the occurrence of the other, i.e. if they cannot occur simultaneously.

For example, a die is tossed. All even outcomes are called events A, while all odd outcomes are called events B. Then A and B are mutually exclusive events that cannot occur simultaneously.

Simple events in a sample space are always mutually exclusive.

Exhaustive events: 

Numerous occurrences are happening in the sample space

As an example, a die is thrown.

Event A has all even outcomes, while Event B has all odd outcomes. Even A and B generate exhaustive events when combined to form a Sample Space.

Axiomatic Approach to Probability:

The Axiomatic approach is another technique for understanding the probability of an event by using axioms or rules. Before assigning probabilities, a few axioms (rules) are predefined in this sort of probability. We may simplify the calculation of event occurrence and nonoccurrence using this way.

Let S represent the sample space for the random experiment. The probability P is a real-valued function with a domain of the power set of S and a range of [0,1] that satisfy the following axioms.

• For any event E,P[E]≥0 
• P[S]=1 
• If E  and F  are mutually exclusive events, then P(E∪F)=P(E)+P(F) 

It follows from (III) that P(ϕ)=0. Let F=ϕ  and  E=ϕ  be two disjointed events, 

∴P(E∪ϕ)=P(E)+P(ϕ) or P(E)=P(E)+P(ϕ)i.e P(ϕ)=0 

Let S  be a sample space containing outcomes ω1,ω2,….ωn i.e., S={ω1,ω2…..,ωn} then 

• 0≤P(ωi)≤1 for each  ω i∈S 
• P(ω1)+P(ω2)+….+P(ωn)=1 
• For any event E,P(E)=∑P(ωi),ωi∈A 
• P(ϕ)=0