## 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 E1 and E2 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 ( E1 ∩ E2 ) = 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 E1 and E2 related to an experiment E, having sample space S are known as mutually exhaustive, if the probability of occurrence of event E1 or E2 is 1.
P ( E1 ∩ E2 ) = 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 E1, E2, E3……..En be n mutually exclusive events related to an experiment, then probability of an event A which occurs with E1 or E2 or E3……..En is given by
P(A) = P(E1) P( A/E1) + P(E2) P( A/E2) + …….. P( A/En)

#### 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