Definition:
Probability means the measurement of confidence.
Suppose a student had the confidence of getting 90 marks out of 100 in tenth maths. Getting 90 is his confidence. His confidence is being measured. That is why the measurement of confidence is called PROBABILITY.
In this explanation, 90 is favourable events and 100 is total events.
Now, we are going to define some important definitions which play a vital role in this chapter.
Random Experiment:
If an experiment is conducted, any number of times, under essentially identical conditions, there is a set of all possible outcomes associated with it.
If the result is not certain and is any one of the several possible outcomes, then the experiment is called a random experiment (or) a trial.
The outcomes are known as elementary events and set of outcomes is an event.
Event:
Event means Result.
Examples:
1) Throwing a die
2) Tossing a coin
3) Picking a card from the pack
Die:
A die is a solid cube which has six faces and numbers 1, 2, 3, 4, 5 and 6 are marked on the faces respectively.
In throwing a die, any one number can be on the uppermost face.
Pack of Cards:
A pack of cards contains 52 cards. Out of these 52 cards, 26 are red coloured and 26 are black coloured.
Again these 52 cards are divided into 4 sets namely Hearts(
), Diamonds
, Spades 
, Clubs 
.
Each set consists of 13 cards, namely A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K. Here A: Ace, K: King, Q: Queen, J: Jack.
Simple Event:
Suppose a die is rolled. To get an even number is an Event. The occurrence of 2 (or) 4 (or) 6 belongs to the event. But the occurrence of say 2, can itself be taken as an event. Thus the event of getting an even number consists of three separate events viz.. getting 2 (or) 4 (or) 6.
Therefore, the event of getting an even number can be split, whereas the event of getting either 2 (or) 4 (or) 6 alone can't be further split. An event in a trial that can not be further split is called a Simple event (or) an Elementary event.
Sample Space:
The set of all possible simple events in a trial is called a sample space. It is denoted by 'S'. Each element of a sample space is called a sample point.
If we toss a coin then
S = { H, T } here H = Head, T = Tail If we toss two coins then
S = { ( H H ), (H T), (T T), (T H) } If we toss three coins then
S = { ( H H H ), ( H H T ), ( H T H ), ( H T T ), ( T T T ), ( T T H),
( T H T ), ( T H H ) } If we toss four coins then
S = { ( H H H H ), ( H H H T), ( H H T T), ( H T T T ),
( T T T H ), ( T T H H ), ( T H H H ), ( T T T T ),
( H H T H ), (T T H T ), ( H T H T ), ( H T T H ), ( T H H T ),
( T H T H ), ( T H T T ), ( H T H H ) } If we throw a die then
S = { 1, 2, 3, 4, 5, 6 } If we throw two dice then
S = { ( 1, 1 ), ( 1, 2 ), ( 1, 3 ), ( 1, 4 ), ( 1, 5 ), ( 1, 6 ),
( 2, 1 ), ( 2, 2 ), ( 2, 3 ), ( 2, 4 ), ( 2, 5 ), ( 2, 6 ),
( 3, 1 ), ( 3, 2 ), ( 3, 3 ), ( 3, 4 ), ( 3, 5 ), ( 3, 6 )
( 4, 1 ), ( 4, 2 ), ( 4, 3 ), ( 4, 4 ), ( 4, 5 ), ( 4, 6 ),
( 5, 1 ), ( 5, 2 ), ( 5, 3 ), ( 5, 4 ), ( 5, 5 ), ( 5, 6 ),
( 6, 1 ), ( 6, 2 ), ( 6, 3 ), ( 6, 4 ), ( 6, 5 ), ( 6, 6 ) }
Equally Likely Events:
Events are said to be equally if they have equal chances to occur.
* When a card is drawn from a pack, any card may be got. In this random experiment, all the 52 elementary events are equally likely.
* When a die is thrown, the elementary events of getting faces numbered 1 to 6 are equally likely.
Exhaustive Events:
All possible events in any random experiment are called exhaustive events.
* In tossing a coin, there are two exhaustive events viz head and tail.
* In throwing of a die, there are six exhaustive events.
Mutually Exclusive Events:
If the happening of one event prevents the happening of other events then they are called mutually exclusive events.
* If a die is thrown, the event of getting an even number and the event of getting an odd number are mutually exclusive events.
* From a pack of cards if a card is drawn, the event of getting a diamond card and the event of getting a club card are mutually exclusive events
Conditional probability:
Suppose A and B are two events of a random experiment.
Then the event "happening of B after the happening of A" is called a conditional event and is denoted by 
Similarly stands for the event "happening of A after the happening of B".
called the Conditional Probability of B after the happening of A, is defined by
Conceptual Theorems
1. State and prove Additional theorem on probability.
Statement: If S is a sample space and E1, E2 are two events then
Proof :
Note:
(1) If E1, E2 are mutually exclusive events then
(2) If E1, E2, ....., En ... are disjoint subsets of S then
(3) Probability of Success = P ( E ) = Probability of happening of an event E.
(4) Probability of failure = P ( E ) = Probability of non-happening of an event E.
(5) 0 ≤ P ( E ) ≤ 1
(6) If P ( E ) = 0 then E is called an impossible event
(7) If P ( E ) = 1 then E is called a certain event.
2. State and prove Multiplication theorem on Probability.
Note:
Before going to state and prove this theorem, students have to define the definition of Conditional Probability. Otherwise full marks will not be given in the examination.
Statement:
If A and B are two events of a random experiment with P ( A ) > 0 and P ( B ) > 0 then
Proof:
Since P ( A ) > 0 and P ( B ) > 0, we have
3. State and prove Baye's theorem.
Statement:
If E1, E2, ..... En are mutually and exhaustive events in a sample space S such
that P ( Ei ) > 0 for i = 1, 2, ... , n and A is any event with P ( A ) > 0 then
Proof:
Since E1, E2 , . . . , En are mutually and exhaustive events in a sample space
S, it follows that 
Ei = S and E1, E2, . . . . . . , En are mutually disjoint.
Now 
are mutually disjoint.
