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Mathematics Grade XI

Grade 11 Mathematics

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Chapter 14: Probability

Class 11 • Mathematics • Chapter 14

Probability

Measuring how likely an event is, on a clear scale from impossible to certain.

CBSE / NCERT Aligned

School Revise

Chapter Roadmap

Random Experiments • Sample Space • Events • Operations on Events • Axiomatic Probability • Addition Theorem • Key Results • Applications

Why Probability Matters

Will it rain tomorrow? What are the chances of winning a game, or of a manufactured part being faulty? Many things in life are uncertain, and probability is the branch of mathematics that measures uncertainty on a clear scale, from 0 for something impossible to 1 for something certain. Everything else sits in between.

In this chapter you will learn to describe an experiment whose result cannot be predicted in advance, to list all its possible outcomes in a sample space, and to talk about events using the language of sets. You will then meet the axiomatic approach, which builds all of probability on a few simple rules, and the addition theorem for finding the chance that one event or another happens.

Key idea

Probability is always a number between 0 and 1. For an experiment with equally likely outcomes, P(event) = (number of favourable outcomes) / (total number of outcomes). An impossible event has probability 0; a certain event has probability 1.

Key Terms You Must Know

Term

Meaning

Example

Random experiment

An action whose result cannot be predicted with certainty but whose outcomes are known.

Tossing a coin

Outcome

A single possible result of an experiment.

Getting a head

Sample space

The set S of all possible outcomes.

For a die, S = {1,2,3,4,5,6}

Event

A subset of the sample space.

Getting an even number, {2,4,6}

Equally likely

Outcomes that each have the same chance.

Each face of a fair die

Mutually exclusive

Two events that cannot happen together (A ∩ B is empty).

Getting a 2 or a 5 on one roll

Complement

The event ‘A does not happen’, written A′.

P(A′) = 1 − P(A)

Core Concepts, Step by Step

1. Random Experiments and Outcomes

A random experiment is one whose result cannot be predicted for sure beforehand, even though we know all the results it could give. Tossing a coin, rolling a die, or drawing a card are all random experiments. Each single result is an outcome: a head, a 4, or the king of hearts. The whole study of probability begins by listing, clearly and completely, all the outcomes an experiment can produce.

2. The Sample Space

The sample space, written S, is the set of all possible outcomes of a random experiment. For one toss of a coin, S = {H, T}; for one roll of a die, S = {1, 2, 3, 4, 5, 6}. For two dice, S has 36 outcomes. The table below shows the sum on two dice for every pair, a sample space of 36 equally likely outcomes that we will use again and again.

The sum on two dice: 36 equally likely outcomes

1st die / 2nd die

3. Events and Their Types

An event is any subset of the sample space, that is, any collection of outcomes we are interested in. Getting an even number on a die is the event {2, 4, 6}. A simple event has just one outcome; a compound event has more than one. The sure event is the whole sample space S (it always happens), and the impossible event is the empty set (it never happens). The complement A′ of an event A is the event that A does not occur.

4. Operations on Events

Because events are sets, we combine them with set operations. The union A ∪ B is the event that A or B (or both) happens. The intersection A ∩ B is the event that both A and B happen. Two events are mutually exclusive when they cannot happen at the same time, so A ∩ B is empty, for example getting a 2 and getting a 5 on a single roll. Events are exhaustive when together they cover the whole sample space.

5. The Axiomatic Approach to Probability

Modern probability rests on three simple rules, or axioms. First, the probability of any event is at least 0. Second, the probability of the sure event S is 1. Third, for mutually exclusive events the probabilities add. From these it follows that every probability lies between 0 and 1. For an experiment whose outcomes are all equally likely, this gives the familiar rule P(A) = (favourable outcomes)/(total outcomes).

6. Complement and the Addition Theorem

Two results are used constantly. The complement rule says P(A′) = 1 − P(A), which is often the quickest route, since the chance that something does not happen is easy to find. The addition theorem says P(A ∪ B) = P(A) + P(B) − P(A ∩ B); we subtract the overlap so it is not counted twice. When A and B are mutually exclusive, the overlap is empty and this simplifies to P(A ∪ B) = P(A) + P(B).

Addition theorem: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Two overlapping events A and B in the sample space

Key Results & Proofs

Three results follow directly from the axioms and are used in almost every probability question.

Result 1: The Complement Rule

Statement. For any event A, P(A′) = 1 − P(A).

Proof

Use the fact that A and A′ together fill the sample space.

An event A and its complement A′ together make up the whole sample space. – A and A′ cover everything

They are mutually exclusive, so their probabilities add. – they cannot both happen

P(A) + P(A′)

=

P(S) = 1

probability of the sure event

P(A′)

=

1 − P(A)

rearrange

So if the chance of rain is 0.3, the chance of no rain is 1 − 0.3 = 0.7.

Result 2: The Addition Theorem

Statement. For any two events, P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

Proof

Think about which outcomes get counted more than once.

Adding P(A) and P(B) counts the overlap A ∩ B twice. – the shared part is double counted

To correct this, subtract the overlap once. – remove one copy

P(A ∪ B)

=

P(A) + P(B) − P(A ∩ B)

the addition theorem

If A and B are mutually exclusive then P(A ∩ B) = 0, giving P(A ∪ B) = P(A) + P(B).

Result 3: Probability of the Impossible Event

Statement. The probability of the impossible event is 0.

Proof

Apply the complement rule to the sure event.

The impossible event is the complement of the sure event S. – it never happens

P(impossible)

=

1 − P(S)

by the complement rule

P(impossible)

=

1 − 1 = 0

since P(S) = 1

So every probability lies between these two extremes: 0 for impossible and 1 for certain.

Worked Examples

Example 1

Question: Write the sample space when a coin is tossed twice.

▶ Show full working

List every possible pair of results in order.

Each toss is a head (H) or a tail (T). – two outcomes per toss

The pairs are HH, HT, TH, TT. – list all combinations

Answer: S = {HH, HT, TH, TT}, four outcomes.

Example 2

Question: A fair die is rolled. Find the probability of getting an even number.

▶ Show full working

Count the favourable outcomes over the total.

Even numbers on a die: {2, 4, 6}, so 3 favourable. – favourable outcomes

P(even)

=

3/6

favourable over total

P(even)

=

1/2

simplify

Answer: 1/2.

Example 3

Question: A fair die is rolled. Find the probability of getting a number greater than 4.

▶ Show full working

The favourable outcomes are 5 and 6.

Numbers greater than 4: {5, 6}, so 2 favourable. – favourable outcomes

P(> 4)

=

2/6 = 1/3

favourable over total

Answer: 1/3.

Example 4

Question: A card is drawn from a well-shuffled pack of 52. Find the probability it is a king.

▶ Show full working

There are 4 kings in a pack.

P(king)

=

4/52

four kings out of fifty-two

P(king)

=

1/13

simplify

Answer: 1/13.

Example 5

Question: A card is drawn from a pack of 52. Find the probability it is a red card.

▶ Show full working

Half the pack is red (hearts and diamonds).

P(red)

=

26/52

twenty-six red cards

P(red)

=

1/2

simplify

Answer: 1/2.

Example 6

Question: Two coins are tossed. Find the probability of getting at least one head.

▶ Show full working

Use the sample space {HH, HT, TH, TT}.

Favourable (at least one head): HH, HT, TH, so 3. – at least one head

P(at least one head)

=

3/4

favourable over total of 4

Answer: 3/4.

Example 7

Question: A die is rolled. Find the probability of not getting a 6.

▶ Show full working

Use the complement rule P(A′) = 1 − P(A).

P(6)

=

1/6

one favourable outcome

P(not 6)

=

1 − 1/6

complement rule

P(not 6)

=

5/6

simplify

Answer: 5/6.

Example 8

Question: A bag holds 5 red and 3 blue balls. One ball is drawn. Find the probability it is red.

▶ Show full working

Count the red balls over the total number of balls.

Total balls = 5 + 3 = 8. – add the balls

P(red)

=

5/8

five red out of eight

Answer: 5/8.

Example 9

Question: If P(A) = 0.5, P(B) = 0.3 and P(A ∩ B) = 0.1, find P(A ∪ B).

▶ Show full working

Use the addition theorem.

P(A ∪ B)

=

P(A) + P(B) − P(A ∩ B)

addition theorem

P(A ∪ B)

=

0.5 + 0.3 − 0.1

substitute

P(A ∪ B)

=

0.7

simplify

Answer: 0.7.

Example 10

Question: A die is rolled. Find the probability of getting a 2 or a 5.

▶ Show full working

These events are mutually exclusive, so the probabilities add.

Getting a 2 and getting a 5 cannot happen together. – mutually exclusive

P(2 or 5)

=

1/6 + 1/6

add the probabilities

P(2 or 5)

=

2/6 = 1/3

simplify

Answer: 1/3.

Example 11

Question: Two dice are thrown. Find the probability that the sum is 7.

▶ Show full working

Count the pairs from the 36 outcomes that add to 7.

Pairs giving 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1), so 6. – favourable pairs

P(sum 7)

=

6/36

favourable over 36

P(sum 7)

=

1/6

simplify

Answer: 1/6.

Example 12

Question: A card is drawn from 52. Find the probability that it is a king or a queen.

▶ Show full working

A card cannot be both, so the events are mutually exclusive.

P(king)

=

4/52

four kings

P(queen)

=

4/52

four queens

P(king or queen)

=

4/52 + 4/52 = 8/52

add (mutually exclusive)

P(king or queen)

=

2/13

simplify

Answer: 2/13.

Where You Meet This in Real Life

Weather forecasts

A ‘70% chance of rain’ is a probability, worked out from past records and current conditions.

Games and sport

Card games, board games and lotteries are built on probability, and selectors use it to weigh a player’s chances.

Insurance

Companies set premiums using the probability of accidents, illness or loss across large groups of people.

Medicine and testing

The chance that a test is correct, or that a treatment works, is described and compared using probability.

Quality and reliability

Factories estimate the probability that a product is faulty so they can keep quality high and waste low.

Practice Sets A to D

Practice Set A – Basics

A1. Write the sample space when one die is rolled.

▶ Reveal full working

List every face.

A die can show 1, 2, 3, 4, 5 or 6.

Answer: S = {1, 2, 3, 4, 5, 6}.

A2. A fair coin is tossed once. Find the probability of a head.

▶ Reveal full working

One favourable outcome out of two.

P(head)

=

1/2

one head out of two outcomes

Answer: 1/2.

A3. A die is rolled. Find the probability of getting a 3.

▶ Reveal full working

One favourable outcome out of six.

P(3)

=

1/6

one favourable out of six

Answer: 1/6.

A4. What is the probability of an impossible event?

▶ Reveal full working

Recall the probability scale.

An impossible event never happens.

Its probability is 0.

Answer: 0.

Practice Set B – Conceptual

B1. What is a sample space?

▶ Reveal full working

Think about all the things that can happen.

It is the set of all possible outcomes of a random experiment.

Answer: The set of all possible outcomes.

B2. What does it mean for two events to be mutually exclusive?

▶ Reveal full working

Can they happen together?

They cannot occur at the same time.

Their intersection A ∩ B is empty.

Answer: They cannot happen together (A ∩ B is empty).

B3. Between what two values does any probability lie?

▶ Reveal full working

Recall the axioms.

No probability is less than 0 or more than 1.

So 0 ≤ P ≤ 1.

Answer: Between 0 and 1 inclusive.

B4. What is the probability of a sure event?

▶ Reveal full working

A sure event always happens.

The sure event is the whole sample space S.

Its probability is 1.

Answer: 1.

Practice Set C – Application / Numerical

C1. A die is rolled. Find the probability of an odd number.

▶ Reveal full working

Odd numbers are 1, 3, 5.

P(odd)

=

3/6 = 1/2

three favourable out of six

Answer: 1/2.

C2. A card is drawn from 52. Find the probability it is a heart.

▶ Reveal full working

There are 13 hearts.

P(heart)

=

13/52 = 1/4

thirteen hearts out of fifty-two

Answer: 1/4.

C3. If P(A) = 0.4, P(B) = 0.5 and P(A ∩ B) = 0.2, find P(A ∪ B).

▶ Reveal full working

Use the addition theorem.

P(A ∪ B)

=

0.4 + 0.5 − 0.2

addition theorem

P(A ∪ B)

=

0.7

simplify

Answer: 0.7.

C4. A bag has 4 red and 6 green balls. One ball is drawn. Find the probability it is green.

▶ Reveal full working

Count green over total.

Total = 4 + 6 = 10.

P(green)

=

6/10 = 3/5

six green out of ten

Answer: 3/5.

Practice Set D – HOTS / Word Problems

D1. Two dice are thrown. Find the probability that the sum is 8.

▶ Reveal full working

Count the pairs from the 36 outcomes that add to 8.

Pairs giving 8: (2,6), (3,5), (4,4), (5,3), (6,2), so 5. – favourable pairs

P(sum 8)

=

5/36

favourable over 36

Answer: 5/36.

D2. A card is drawn from 52. Find the probability it is a face card (jack, queen or king).

▶ Reveal full working

There are 12 face cards in all.

Face cards: 3 in each of 4 suits, so 12. – count the face cards

P(face card)

=

12/52 = 3/13

twelve out of fifty-two

Answer: 3/13.

D3. A die is rolled. Find the probability of getting a prime number.

▶ Reveal full working

The primes on a die are 2, 3 and 5.

Prime numbers on a die: {2, 3, 5}, so 3. – favourable outcomes

P(prime)

=

3/6 = 1/2

three favourable out of six

Answer: 1/2.

D4. The probability that a student passes an exam is 0.85. Find the probability that the student does not pass.

▶ Reveal full working

Use the complement rule.

P(not pass)

=

1 − P(pass)

complement rule

P(not pass)

=

1 − 0.85 = 0.15

substitute

Answer: 0.15.

Chapter Summary

Everything in One Glance

Random Experiment

An action whose result is uncertain but whose possible outcomes are known.

Sample Space

The set S of all possible outcomes; an event is any subset of S.

Events

Simple, compound, sure (S), impossible (empty), and complement A′.

Equally Likely

P(A) = (favourable outcomes)/(total outcomes), and 0 ≤ P(A) ≤ 1.

Complement Rule

P(A′) = 1 − P(A); the impossible event has probability 0, the sure event 1.

Addition Theorem

P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for mutually exclusive events, just P(A) + P(B).

Are You Exam-Ready?

8-Point Exam Quick-Check

Write the sample space for tossing two coins.

A die is rolled; find P(getting a number less than 3).

A card is drawn from 52; find P(it is a spade).

State the range of values a probability can take.

If P(A) = 0.2, find P(A′).

Two dice are thrown; find P(sum = 6).

If P(A) = 0.3, P(B) = 0.4 and the events are mutually exclusive, find P(A ∪ B).

What is the probability of the sure event?

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Class 11 Mathematics Chapter 14: Probability, Complete Notes and Practice

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