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Probability Class 11th Notes - Free NCERT Class 11 Maths Chapter 16 notes - Download PDF

Probability Class 11th Notes - Free NCERT Class 11 Maths Chapter 16 notes - Download PDF

Edited By Komal Miglani | Updated on Apr 11, 2025 01:38 PM IST

Probability is a way to measure how likely something is to happen. For example, when you toss a coin, there’s an equal chance of getting heads or tails. So, the probability of getting heads is 1 out of 2. Probability is always a number between 0 and 1, both 0 and 1 inclusive. If something can’t happen, its probability is 0. If it will surely happen, its probability is 1. We use probability in real life to make guesses about things like the weather, winning a game, or making decisions. It helps us understand and handle situations where the result is not certain. Class 11 Maths Chapter 14 notes contain the following topics: sample space, outcomes, types of events, mutually exclusive events, exhaustive events, axiomatic approach, Addition rule, etc.

This Story also Contains
  1. NCERT CLASS 11 CHAPTER 14 NOTES
  2. Importance of NCERT Class 11 Maths Chapter 16 Notes
  3. NCERT Class 11 Notes Chapter Wise
  4. Subject Wise NCERT Solutions

Class 11 Maths chapter 14 notes are prepared by subject matter experts in a precise manner so that students can get maximum benefit from them. The notes for Class 11 Maths Chapter 14 help a student to get a last-minute revision before the exam. Notes for Class 11 Maths Chapter 14 are made in such a way that no students face any difficulty during their preparation. NCERT Notes for Class 11 Maths cover the entire NCERT syllabus for this chapter.

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NCERT CLASS 11 CHAPTER 14 NOTES

Random Experiment:

An experiment is random means that the experiment has more than one possible outcome, and it is not possible to predict with certainty which outcome will be. For instance, in an experiment of tossing an ordinary coin, it can be predicted with certainty that the coin will land either heads up or tails up, but it is not known for sure whether heads or tails will occur. If a die is thrown once, any of the six numbers, i.e., 1,2,3,4,5,6 may turn up, not sure which number will come up.
(i) Outcome A possible result of a random experiment is called its outcome for example if the experiment consists of tossing a coin twice, some of the outcomes are HH,HT, etc.
(ii) Sample Space A sample space is the set of all possible outcomes of an experiment. In fact, it is the universal set S pertinent to a given experiment.
The sample space for the experiment of tossing a coin twice is given by

S={HH,HT,TH,TT}

The sample space for the experiment of drawing a card out of a deck is the set of all cards in the deck.

Outcomes:

The solution of the experiment is called the outcome.

Sample Space:

The set of all possible chances or outcomes is called the sample space.

Event:

An event is a subset of a sample space S. For example, the event of drawing an ace from a deck is

A={ Ace of Heart, Ace of Club, Ace of Diamond, Ace of Spade }

Event ' A or B ': If A and B are two events associated with the same sample space, then the event 'A or B' is the same as the event A∪B and contains all those elements which are either in A or in B or both. Furthermore, P(A∪B) denotes the probability that A or B (or both) will occur.
Event ' A and B ': If A and B are two events associated with a sample space, then the event ' A and B ' is the same as the event A∩B and contains all those elements which are common to both A and B. Further more, P(A∩B) denotes the probability that both A and B will simultaneously occur.
The Event 'A but not B ': (Difference A−B ) An event A−B is the set of all those elements of the same space S which are in A but not in B, i.e., A−B=A∩B′.
Mutually exclusive: Two events A and B of a sample space S are mutually exclusive if the occurrence of any one of them excludes the occurrence of the other event. Hence, the two events A and B cannot occur simultaneously, and thus P(A∩B)=0.

Exhaustive events: If E1,E2,…,En are n events of a sample space S and if

E1∪E2∪E3∪…∪En=⋃i=1nEi=S

then E1,E2,…,En are called exhaustive events.
In other words, events E1,E2,…,En of a sample space S are said to be exhaustive if at least one of them necessarily occur whenever the experiment is performed.
Consider the example of rolling a die. We have S={1,2,3,4,5,6}. Define the two events

A: 'a number less than or equal to 4 appears.'
B 'a number greater than or equal to 4 appears.'

Now
A:{1,2,3,4},B={4,5,6}
A∪B={1,2,3,4,5,6}=S
Such events A and B are called exhaustive events.

Mutually exclusive and exhaustive events: If E1,E2,…,En are n events of a sample space S and if Ei∩Ej=ϕ for every i≠j, i.e., Ei and Ej are pairwise disjoint and ⋃i=1nEi=S, then the events E1,E2,…,En are called mutually exclusive and exhaustive events.

Consider the example of rolling a die.
We have S={1,2,3,4,5,6}
Let us define the three events as
A=a number which is a perfect square
B= a prime number
C= a number which is greater than or equal to 6
Now A={1,4},B={2,3,5},C={6}
Note that A∪B∪C={1,2,3,4,5,6}=S. Therefore, A,B, and C are exhaustive events.
Also A∩B=B∩C=C∩A=ϕ
Hence, the events are pairwise disjoint and thus mutually exclusive.
The classical approach is useful when all the outcomes of the experiment are equally likely. We can use logic to assign probabilities. To understand the classical method consider the experiment of tossing a fair coin. Here, there are two equally likely outcomes - head (H) and tail (T). When the elementary outcomes are taken as equally likely, we have a uniform probability model. If there are k elementary outcomes in S, each is assigned the probability of 1k. Therefore, logic suggests that the probability of observing a head, denoted by P(H), is 12=0.5, and that the probability of observing a tail, denoted P(T), is also 12=5. Notice that each probability is between 0 and. Further, H and T are all the outcomes of the experiment and P(H)+P(T)=1.

Events are of different types:

(i) Impossible and Sure Events: The empty set ϕ and the sample space S describe events. In fact, ϕ is called an impossible event and S, i.e., the whole sample space is called a sure event.

(ii) Simple or Elementary Event: If an event E has only one sample point of a sample space, i.e., a single outcome of an experiment, it is called a simple or elementary event. The sample space of the experiment of tossing two coins is given by

S={HH,HT,TH,TT}

The event E1={HH} containing a single outcome HH of the sample space S is simple. If one card is drawn from a well-shuffled deck, any particular card drawn like 'Queen of Hearts' is an elementary event.
(iii) Compound Event: If an event has more than one sample point it is called a compound event, for example, S={HH,HT} is a compound event.
(iv) Complementary event: Given an event A, the complement of A is the event consisting of all sample space outcomes that do not correspond to the occurrence of A.

The complement of A is denoted by A ' or A¯. It is also called the event 'not A '. Further P(A―) denotes the probability that A will not occur.

A′=A―=S−A={w:w∈ S and w∉ A}

Axiomatic Approach To Probability:

Let S be the sample space of a random experiment. The probability P is a real-valued function whose domain is the power set of S, i.e., P(S) and range is the interval [0,1] i.e. P:P(S)→[0,1] satisfying the following axioms.
(i) For any event E,P(E)≥0.
(ii) P(S)=1
(iii) 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 S be a sample space containing elementary outcomes w1,w2,…,wn, i.e., S={w1,w2,…,wn}

It follows from the axiomatic definition of probability that
(i) 0≤P(wi)≤1 for each wi∈ S
(ii) P(wi)+P(w2)+…+P(wn)=1
(iii) P(A)=P(wi) for any event A containing elementary events wi

For example, if a fair coin is tossed once

P(H)=P(T)=12 satisfies the three axioms of probability.

Now suppose the coin is not fair and has double the chances of falling heads up as compared to the tails, then P(H)=23 and P(T)=13.
This assignment of probabilities is also valid for H and T as these satisfy the axiomatic definitions.

Probability of An Event:

S be the sample space, E be the event, and n(s)= n and n(E)=m

Then the probability of the event= P(E)

P(E)= Number of favorable outcomes total number of possible outcomes =mn

Odd number of outcomes in favor of the event: m : (n-m)

Odd number of outcomes against the event : (n-m): m

Probability of the event that does not occur or take place : P(A)= 1- P(A)

Probabilities of equally likely outcomes

Let a sample space of an experiment be S={w1,w2,…,wn} and suppose that all the outcomes are equally likely to occur i.e., the chance of occurrence of each simple event must be the same

i.e., P(wi)=p for all wi∈ S, where 0≤p≤1 Since nP(wi)=1 i.e., p+p+p+…+p(n times )=1⇒np=1, i.e. p=1n

Let S be the sample space and E be an event, such that n( S)=n and n(E)=m. If each outcome is equally likely, then it follows that

P(E)=mn= Number of outcomes favourable to E Total number of possible outcomes

Addition Rule :

Probability of event A or B :

P(A∪B) = P(A) + P(B) - P(A∩B)

Probability of event A or B or C :

P(A∪B∪C) = P(A) + P(B) +P(C)-P(A∩B)-P(B∩C) - P(A∩C) + P(A∩B∩C)

If A, B are mutually exclusive events: Then P(A∪B) = P(A) + P(B)

Since A∩B in an exclusive event is zero or null.

Addition rule for mutually exclusive events

If A and B are disjoint sets, then P(A∪B)=P(A)+P(B) [since P(A∩B)=P(ϕ)=0, where A and B are disjoint]. The addition rule for mutually exclusive events can be extended to more than two events.

With this topic, we conclude NCERT Class 11 chapter 16 notes.

Importance of NCERT Class 11 Maths Chapter 16 Notes

NCERT Class 11 Maths Chapter 16 notes will be very helpful for students to score good marks in their 11 class exams. Students can download Probability Class 11 chapter 16 pdf to revise the concepts in short period of time before the exam.

  • NCERT problems are very important in order to perform well in exams. Students must try to solve all the NCERT problems, including miscellaneous exercises, and if needed, refer to the NCERT Solutions for Class 11 Maths Chapter 16 Probability.
  • Students are advised to go through the NCERT Class 11 Maths Chapter 16 Notes before solving the questions.
  • To boost your exam preparation as well as for quick revision, these NCERT notes are very useful.
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A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

Option 1)

0.34\; J

Option 2)

0.16\; J

Option 3)

1.00\; J

Option 4)

0.67\; J

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times.  Assume that the potential energy lost each time he lowers the mass is dissipated.  How much fat will he use up considering the work done only when the weight is lifted up ?  Fat supplies 3.8×107 J of energy per kg which is converted to mechanical energy with a 20% efficiency rate.  Take g = 9.8 ms−2 :

Option 1)

2.45×10−3 kg

Option 2)

 6.45×10−3 kg

Option 3)

 9.89×10−3 kg

Option 4)

12.89×10−3 kg

 

An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

Option 1)

2,000 \; J - 5,000\; J

Option 2)

200 \, \, J - 500 \, \, J

Option 3)

2\times 10^{5}J-3\times 10^{5}J

Option 4)

20,000 \, \, J - 50,000 \, \, J

A particle is projected at 600   to the horizontal with a kinetic energy K. The kinetic energy at the highest point

Option 1)

K/2\,

Option 2)

\; K\;

Option 3)

zero\;

Option 4)

K/4

In the reaction,

2Al_{(s)}+6HCL_{(aq)}\rightarrow 2Al^{3+}\, _{(aq)}+6Cl^{-}\, _{(aq)}+3H_{2(g)}

Option 1)

11.2\, L\, H_{2(g)}  at STP  is produced for every mole HCL_{(aq)}  consumed

Option 2)

6L\, HCl_{(aq)}  is consumed for ever 3L\, H_{2(g)}      produced

Option 3)

33.6 L\, H_{2(g)} is produced regardless of temperature and pressure for every mole Al that reacts

Option 4)

67.2\, L\, H_{2(g)} at STP is produced for every mole Al that reacts .

How many moles of magnesium phosphate, Mg_{3}(PO_{4})_{2} will contain 0.25 mole of oxygen atoms?

Option 1)

0.02

Option 2)

3.125 × 10-2

Option 3)

1.25 × 10-2

Option 4)

2.5 × 10-2

If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

Option 1)

decrease twice

Option 2)

increase two fold

Option 3)

remain unchanged

Option 4)

be a function of the molecular mass of the substance.

With increase of temperature, which of these changes?

Option 1)

Molality

Option 2)

Weight fraction of solute

Option 3)

Fraction of solute present in water

Option 4)

Mole fraction.

Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol-1) is

Option 1)

twice that in 60 g carbon

Option 2)

6.023 × 1022

Option 3)

half that in 8 g He

Option 4)

558.5 × 6.023 × 1023

A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t2) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m2 , the number of rotations made by the pulley before its direction of motion if reversed, is

Option 1)

less than 3

Option 2)

more than 3 but less than 6

Option 3)

more than 6 but less than 9

Option 4)

more than 9

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