## Probability

Probability means the event that something will happen.** Probability happening of event = Number of favourable outcomes /Total number of possible outcomes**

P(E) = n(E)/n(s)

Where, P(E) = Probability of an event

n(E) = number of favourable outcomes

n(S) = total number of possible outcomes

The probability of any event (E) is a number between 0 and 1.

0 ≤ P(E) ≤ 1

**Ex**- There are 7 balls in a bag. 4 of them are red and 3 of them are blue. What is the probability that a blue ball will be picked?**Solution**:- given, number of favourable outcomes = 3

Total number of outcomes = 7

Required probability = Number of favourable outcomes /Total number of possible outcomes

= 3/7

### Terms Related to Probability

#### Experiment

An action or operation resulting in two or more outcomes is called an experiment.

For example- Tossing of a coin, Throwing a coin.

#### Sample Space

The set of all possible outcomes of an experiment is called the sample space, denoted by S.

For example- If we throw a die, then sample space

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

If we toss a coin, then sample space S = { H, T }

#### Possible outcomes

All possibilities related to an event are known as possible outcomes.

For example- When a coin is tossed, these are two possible outcomes, Probability of getting H = 1/2

Probability of getting T = 1/2.

When a single die is thrown, there are six possible outcomes 1, 2, 3, 4, 5, 6. Probability of getting any one of these number is 1/6.

#### Event

Any subset of a sample space is an event.

For example- Getting a head is an event related to tossing of a coin.

### Impossible Event

Impossible event has no chance of occurring. the null set ( φ ) is called impossible event or null event.

For example- Getting 7 when a dice is thrown is an impossible or a null event.

**Ex**- A bag contains 15 black balls, if a ball is picked at random from the bag. Find the probability that ball picked is of Red color.**Solution**:- The bag contains 15 black balls and there is no red ball in the bag.

So, number of favorable even = 0

Total outcomes = 15

∴ Probability = 0/15 = 0

#### Sure Event

The entire sample space is called sure or certain event.

For example- Getting an odd or event number on throwing a dice is a sure event.

#### Equally Likely Events

A number of simple events are said to be equally likely if there is no reason for one event to occur in preference to any other event.

For example- When a dice is rolled the possible outcome of getting an odd number = 3, possible outcome of getting an even number = 3

So, getting a even number or odd number are equally likely events.

#### Compound Event

It is joint occurrence of two or more simple events

For example- The event of at least one head appears when two fair coins are tossed is a compound event, A{ HT, TH, HH }

#### Mutually Exclusive

Two events E_{1} and E_{2} related to an experiment E, having sample space S are known as mutually exclusive.

If the probability of occurrence of both events simultaneously is zero.

P ( E_{1} ∩ E_{2} ) = 0

For example- When a coin is tossed either head or tail will appear. Head and tail coin not occur simultaneously. Therefor occurrence of a head or a tail are two mutually exclusive events

#### Exhaustive Events

Two events E_{1} and E_{2} related to an experiment E, having sample space S are known as mutually exhaustive, if the probability of occurrence of event E^{1} or E^{2} is 1.

P ( E_{1} ∩ E_{2} ) = 1

For example-Let A be probability of getting an even number when a dice is rolled and B be the probability of getting an odd number. The probability the probability of occurrence of event A or B is i.e., any of the even can occur so they are mutually exhanstive.

#### Dependent Events

If the outcome or occurrence of the first affects the outcome or occurrence of the second, then two events are called dependent.

#### Independent Events

Two or more events are said to be independent if occurrence or non-occurrence of any of them does not influence the probability of occurrence or non-occurrence of other events.

If A and B are independent events, then

P( A and B ) = P( A ∩ B ) = P(A) . P(B)

For example- Getting head after tossing a coin and getting a 5 on a rolling single 6-sided die are independent events.

**Ex**- A fair coin is tossed repeatedly. If the tail appears on first four tosses, then the probability of the head appearing on the fifth toss equals.**Solution**:- The event that the fifth toss results a head is independent of the event that the first four tosses results tails.

∴ Probability of the required event = 1/2.

#### Law of Total Probability

If E_{1}, E_{2}, E_{3}……..E_{n} be n mutually exclusive events related to an experiment, then probability of an event A which occurs with E_{1} or E_{2} or E_{3}……..E_{n} is given by

**P(A) = P(E**_{1}**) P( A/E**_{1}**) + P(E**_{2}**) P( A/E**_{2}**) + …….. P( A/E**_{n}**)**

#### Conditional Probability

The conditional probability of an event B in relationship to an event A is the probability that event B occurs given that event A has already been occurred. The notation for conditional probability is P (B/A). It is pronounced as the probability of happening of an event B given that A has already been happened.

P(A/B) = P( A ∩ B )/ P(B) and

P(B/A) = P( A ∩ B )/ P(A)

**Ex**- A English teacher conducted two tests in her class. 35% of the students passed both tests and 70% of the students passed the first test . What percent of the students passed the second test given that they have already passed the first test?**Solution**:-

P(B/A) = P( A and B )/ P(A)

= 35/70

= 1/2

= 50%

### Rules Related to Probability

#### Addition Rule of Probability

When two events A and B are mutually exclusive, the probability that A or B will occur, is the sum of the probability of each event.

P ( A or B ) = P(A) + P(B) and P( A ∩ B ) = P(A) + P(B)

But when two events A and B are non-mutually exclusive, the probability that A or b will occur, is

P ( A or B ) = P(A) + P(B) - P( A and B )

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

Ex- From a well shuffled pack of 52 cards, a card is frawer at random, find the probability that it is either a heart or a queen.

Solution:- Let A be the probability of getting a heart card and B be the probability of getting a queen card

P(A) = 13/52 P(B) = 4/52 P( A ∩ B ) = 1/52

∴ Required probability = P ( A ∪ B ) = P(A) + P(B) - P( A ∩ B )

= 13/52 + 4/52 - 1/52

= 16/52 = 4/13

#### Multiplication Theorem of Probability

When two events A and B are mutually exclusive, the probability that A and B will occur simultaneously is given as

P ( A ∩ B ) = P(A) . P(B) ( A and B are independent event )

**Ex**- Four persons P, Q, R and S appeared for an interview. Find the probability that both P and S are selected in the interview?**Solution**:-

Probability that P is selected = P (P) = 1/4

Probability that S is selected = P (S) = 1/4

∴ Required probability that both are selected = 1/4 × 1/4 = 1/16

### Types of Questions

#### Question Based on Coins

This types of question are based on tossing of coins and obtaining a particular face (Head/Tail) or obtaining same face on two or more coins.

**Ex**- If a coin is tossed what is the probability of each outcome?**Solution**:- Here, total outcomes = ( H, T ) = 2

favourable outcome = 1

Required probability = 1/2

**Ex**- If a coin is tossed twice, then find the probability that a Tail is obtained atleast once.**Solution**:- Total outcomes = ( HH, TT, HT, TH ) = 4

Favourable outcome = ( TT, HT, TH ) = 3

∴ Required probability = 3/4

#### Question Based on Dice

This type of questions are based on rolling of one or more dice and getting a particular number on the face or a particular sum on faces of the dice etc.

**Ex**- A single 6-sided die is rolled. What is the probability of getting prime number and getting composite number?**Solution**:- Total outcomes = ( 1, 2, 3, 4, 5, 6 ) = 6

Prime numbers = ( 2, 3, 5 ) = 3

Composite number = ( 4, 6 ) = 2

Probability of getting prime number = 3/6 = 1/2

Probability of getting composite number = 2/6 = 1/3

**Ex**- A single 6-sided die is rolled. What is the probability of getting a 3 or a 6?**Solution**:- Total outcomes = ( 1, 2, 3, 4, 5, 6 ) = 6

Probability of getting a number in a single throw of die = 1/6

Probability of getting 2 = 1/6

Probability of getting 6 = 1/6

∴ Required probability of getting either 3 or 6 = 1/6 + 1/6 = 2/6 = 1/3

#### Question Based on Playing Cards

There are total 52 cards in a deck of playing cards.

There are 26 red and 26 black cards.

There are 13 cards of each suit Clubs, Diamonds, Hearts and Spades.

There are 4 Aces, 4 Jacks, 4 Queens and 4 Kings.

There are 12 face cards.

**Ex**- A total of six cards are chosen at random from a standard deck of 52 playing cards. What is the probability of choosing 6 kings?**Solution**:- There are only 4 kings, so we cannot choose 6 kings.

Thus, it is impossible event.

∴ Probability = 0/52 = 0

**Ex**- A single card is chosen at random from a standard deck of 52 playing cards. What is the probability of choosing a jack or a diamond?**Solution**:- There are 4 jack in a standard deck and 13 diamond.

Also 1 jack is of diamond. so probability of getting a jack = 4/52

Probability of getting a jack of diamond = 13/52

Probability of getting a jack of diamond = 1/52

∴ Required probability of getting a jack or diamond

= 4/52 + 13/52 - 1/52 = 16/52 = 4/13

#### Question Based on Marbles or Balls

This types of questions are based on choosing a ball or a marble of particular color from one or more bag containing different colored balls or marbles.

**Ex**- A bag contains 3 red, 4 green and 5 black balls. If a single ball is chosen at random from the jar, what is the probability that it is red or black?**Solution**:- Total outcomes = 3 + 4 + 5 = 12

Probability of getting a red ball = 3/12

Probability of getting a black ball = 5/12

∴ Probability of getting red or black = 3/12 + 5/12 = 8/12 = 2/3