Probability

When a coin is tossed we cannot know whether it shows heads or tails, but we can say both have an equal chance. We put this “chance” into a number: the probability. If all outcomes are equally likely, probability = favourable outcomes ÷ total outcomes. So probability always lies between 0 and 1: 0 means the event is impossible, 1 means it is certain. The whole chapter is practice in counting this fraction correctly in different settings — coins, dice, cards, marbles — and then the rules for combining events.

P (E) = \frac{n ( E )}{n ( S )}, 0 \leq P (E) \leq 1, P (\overset{ˉ}{E}) = 1 - P (E)

Three basics

1

Basic ideas

Sample space, event, complement

All possible outcomes, favourable outcomes, certain and impossible events, and the complement.

Terms and simple questions

Favourable ÷ total

Experiment, sample space $S$ , event $E$ and complement $\overset{ˉ}{E}$ .

Term	Meaning
Experiment	An action whose possible outcomes are known but whose result is not (tossing a coin)
Sample space $S$	The set of all possible outcomes; for a die ${1, 2, 3, 4, 5, 6}$
Event $E$	Any part of $S$ — like “an even number comes”
Certain / impossible	$P (S) = 1$ / $P (\emptyset) = 0$
Complement $\overset{ˉ}{E}$	The event that $E$ does not happen; $P (\overset{ˉ}{E}) = 1 - P (E)$

Example 1 · Basic

A bag holds 3 red and 2 blue marbles. What is the probability that a marble drawn at random is blue?

Solution, step by step

1Total marbles $= 5$ ; favourable (blue) $= 2$ .
2Probability $= \frac{2}{5}$

Answer $\frac{2}{5}$

Example 2 · Basic

A card is drawn at random from 52. What is the probability that it is not a king?

Solution, step by step

1P(king) $= \frac{4}{52} = \frac{1}{13}$ .
2Not a king $= 1 - \frac{1}{13} = \frac{12}{13}$

Answer $\frac{12}{13}$

BasicTry it yourself first

A bag holds 4 black, 6 white marbles. One marble is drawn at random. What is the probability that it is black?

BasicTry it yourself first

A bag holds 7 black, 5 white marbles. One marble is drawn at random. What is the probability that it is white?

StandardTry it yourself first

A bag holds 5 red, 3 green, 4 blue, 8 yellow marbles. One marble is drawn at random. What is the probability that it is yellow or green?

StandardTry it yourself first

A bag holds 2 red, 6 green, 5 blue, 7 yellow marbles. One marble is drawn at random. What is the probability that it is red or blue?

AdvancedTry it yourself first

A bag holds 9 red, 4 green, 6 blue, 11 yellow marbles. One marble is drawn at random. What is the probability that it is green or yellow?

AdvancedTry it yourself first

A bag holds 8 red, 5 blue, 7 green marbles. One marble is drawn at random. What is the probability that it is blue or green?

BasicTry it yourself first

A card is drawn at random from a pack of 52. What is the probability that it is not a king?

BasicTry it yourself first

A card is drawn at random from a pack of 52. What is the probability that it is not a red card?

StandardTry it yourself first

A card is drawn at random from a pack of 52. What is the probability that it is not a face card?

StandardTry it yourself first

A card is drawn at random from a pack of 52. What is the probability that it is not a spade?

StandardTry it yourself first

A card is drawn at random from a pack of 52. What is the probability that it is not an ace?

BasicTry it yourself first

A letter is chosen at random from the word 'EDUCATION'. What is the probability that it is a vowel?

StandardTry it yourself first

A letter is chosen at random from the word 'MATHEMATICS'. What is the probability that it is a vowel?

StandardTry it yourself first

A letter is chosen at random from the word 'ELEPHANT'. What is the probability that it is a consonant?

AdvancedTry it yourself first

A letter is chosen at random from the word 'MOUNTAIN'. What is the probability that it is a vowel?

AdvancedTry it yourself first

A letter is chosen at random from the word 'CHEMISTRY'. What is the probability that it is a consonant?

StandardTry it yourself first

What is the probability that a randomly chosen leap year has 53 Sundays?

StandardTry it yourself first

What is the probability that a randomly chosen non-leap year has 53 Sundays?

AdvancedTry it yourself first

What is the probability that a randomly chosen leap year has 53 Sundays?

StandardTry it yourself first

A number is chosen at random from 1, 2, ..., 40. What is the probability that it is a prime?

StandardTry it yourself first

A number is chosen at random from 1, 2, ..., 30. What is the probability that it is a multiple of 4?

StandardTry it yourself first

A number is chosen at random from 1, 2, ..., 60. What is the probability that it is a perfect square?

AdvancedTry it yourself first

A number is chosen at random from 2, 3, ..., 50. What is the probability that it is a number with units digit 7?

AdvancedTry it yourself first

A number is chosen at random from 1, 2, ..., 100. What is the probability that it is divisible by both 2 and 3?

2

Coins, dice, cards and marbles

Four familiar experiments

The sample space of each experiment and combinations for multiple draws.

Coins

2^{n}

outcomes

$n$ coins have $2^{n}$ outcomes; exactly $k$ heads in $^{n} C_{k}$ ways.

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Dice

6 or 36 outcomes

A die has 6 outcomes and two dice have 36.

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Playing cards

The pack of 52

4 suits (spades, hearts, diamonds, clubs) $\times$ 13 cards; 26 red, 26 black; 12 face cards.

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Marbles and balls

Count with combinations

When several marbles are drawn together favourable and total ways come from $^{n} C_{r}$ .

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3

Addition and multiplication rules

“Or” and “and”

$P (A \cup B)$ , independent events $P (A) P (B)$ , conditional probability and total probability.

The addition rule

P (A \cup B)

The probability of “A or B”; for mutually exclusive events the overlap is zero.

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Multiplication and independent events

P (A \cap B) = P (A) P (B)

Independent events, the “at least one” rule, the binomial and draws without replacement.

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Conditional probability

P (B / A) = \frac{P ( A \cap B )}{P ( A )}

The probability of $B$ given that $A$ has already happened.

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4

Selection, rows and pigeonhole

Special cases

Selecting two particular persons, sitting next to each other and certain-event questions like socks.

Special questions

Counting formulas inside probability

Use permutations and combinations for the favourable and the total counts.

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5

Traps and fast tricks

Slips · tricks

Where marks go in probability questions.

The commonest mistakes

Six traps

The slips that cost marks.

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Fast tricks

Complement and listing

Habits for speed.

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6

The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct rules

One-rule questions.

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Level 2 — Standard

Two ideas combined

Exam-level questions.

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Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

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7

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.

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Basic ideas

Terms and simple questions

Coins, dice, cards and marbles

Coins

Dice

Playing cards

Marbles and balls

Addition and multiplication rules

The addition rule

Multiplication and independent events

Conditional probability

Selection, rows and pigeonhole

Special questions

Traps and fast tricks

The commonest mistakes

Fast tricks

The practice ladder

Level 1 — Basic

Level 2 — Standard

Level 3 — Advanced

Revision — formula cards

Free All Access Passevery mind map · Hindi + English · 12 months

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