Probability.

Probability.

The possibility of happening or not happening of an event is known as probability.

Probability.
The possibility of the happening of an event = (Number of favorable outcomes / Total number of outcomes)

Experiment
An operation which can produce some well-defined outcomes is called an experiment.

Random Experiment:
An experiment in which all possible outcomes are known and the exact output cannot be predicted in advance, is called a
random experiment.

Examples of Performing a Random Experiment :
1. Rolling an unbiased dice.
2. Tossing a fair coin.
3. Drawing a card from a pack of well-shuffled cards.
4. Picking up a ball of a certain color from a bag containing balls of different colors.

Details:
1. When we throw a coin, then either a Head (H) or a Tail (T) appears.
2. A dice is a solid cube, having 6 faces, marked 1, 2, 3, 4, 5, 6 respectively. When we throw a die, the outcome is the number
that appears on its upper face.
3.A pack of cards has 52 cards.
4.It has 13 cards of each suit name Spades, Clubs, Hearts and Diamonds.
5.Cards of spades and clubs are black cards.
6.Cards of hearts and diamonds are red cards.

7.There are 4 honours of each unit.

8.There are Kings, Queens and Jacks. These are all called face 

cards

Event
Any subset of a sample space is called an event.

Probability of Occurrence of an Event:

Let S be the sample and let E be an event.
Then, E ⊆ S

P( E )= n(E)/n(S)

Results on Probability :
P(s) = 1

0≤P (E)≤1

P(Ø)=0
For any events A and B we have : 

P( A U B)=P(A) + P(B) –P(A⋂B)

Sample space

When we perform an experiment, then the set S of all possible outcomes is called the sample space. It is denoted by S.
When two dice are thrown simultaneously, then
Sample space (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) }
Thus, The sample space contains 36 sample points.

Shortcut Tricks
When two dice are thrown The probability of sample points for the given sum can be known with below logic.

Total Number of sample points when two dice are thrown = (6* 6 )=36

Examples:
In tossing a coin, S = {H, T}
If two coins are tossed, then S = { HH,HT,TH, TT }
In rolling a dice, we have, S = {1, 2, 3, 4,5,6}

Basic Formulas

P(E) = n(E)/n(S)

Example sum
Two dice are thrown together. What is the probability that the sum of the number on the two faces is divided by 4?

N(S)=6*6
= 36
Let E be the event that the sum of the numbers on the two faces is divided by 4.

E={ (1,3), (2,2),(2,6),(3,1),(3,5),(4,4),(5,3) ,(6,2)}

n (E)=8

p(E)= n(E)/n(S)

=8/36

=2/9