Basic Theory of Probability Chapter(i) 
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Probability is a study of mathematics
about chances. When things
happen randomly method of probability is used to find how likely a
particular event will occur. Just imagine a coin being tossed up in the air and
the gravity of earth will bring it back to the ground and now the coin will land on
its one of the flat surfaces, but we don't know for sure which side of the flat
surfaces of it is going to be in contact with the ground or which side of it is
going to be visible to us, either
head or tail in both ways. Nevertheless, as a matter of fact there are equal chances for head or tail
landing face up provided that the coin is in pristine condition. What we all know for sure is that there are only two ways the coin
would have landed either with head facing up or tail facing up. Since these two
outcomes are the only possible total number of outcomes, we call these outcomes
as members of the SAMPLE SPACE, which
consist of all possible outcomes from an occurrence.
(Someone could think there are also
possible other two outcomes at which, it could have landed on its curved
surface standing straight up, or it could not have fallen
back at all by escaping
from the pull of gravity. For the sake of argument both possibilities are valid and
countable. However, these two outcomes have a very remote chance to happen,
either
of which, we call a Zero Probability,
which we will discuss little later.)
Despite above all being said about the probability I just want to mention some thing to rectify the wrong concept that some readers might have about the study of probability. Probability can't predict anything. However, another branch of mathematics known as statistics employs probability along with other several methods of its own to predict future outcomes like for an example, behavior of stock market or unemployment rate in the future based upon the data obtained from various sources.
Probability
Experiments and Sample Space: A sample space is populated with outcomes
obtained from repeated
probability experiments.
Example; tossing a coin several time and writing down
the outcomes. An outcome is a result obtained from a test trial. Following table
shows a sample space from two different probability experiments coin tossing
and dice rolling.
Experiment  Sample Space 
Toss of a coin  Head (H), Tail (T) 
Roll of a die  1, 2, 3, 4, 5, 6 
Or you can represent them in a Venn Diagram, since outcomes of an occurrence can be considered a set.
Each outcome from the sample space is called a member or an element.
Classic
Probability:
Let's come back to the discussion of coin tossing, and an outcome, at which our
coin landed showing its face up. We call this is an event of getting a
head from coin toss and obviously the other event is getting a tail.
Therefore,
the Sample Space from Coin tossing is
Probability to get a head is 1/2, because probability is defined by this following formula
P (E) = 

Where P(E) stands for the "Probability of an event". To calculate the probability of getting a tail from the coin toss, when a coin is tossed as we discussed there are two outcomes, now look at our sample space, which is {H,T} where one chance to get a tail while total outcomes are two, now by applying the probability formula.
P (Tail) = 

Multiple coin tossing: When two different coins are tossed simultaneously as you can expect there are total of 4 outcomes. Our sample space in this case will be {HH, HT, TH, TT}
Number of possible outcomes can also be mathematically computed without preparing a
sample space, which is 2x2 = 4. Likewise when 3 coins are tossed number of
total outcomes will be 2x2x2 = 8. If n coins are tossed total members in the
sample space will be 2^{n}. That is the total number of outcomes when n
coins are tossed.
Rolling a Die:
As we know a die (singular form of dice) is a cubic solid object with six faces each
of which numbered from 1 to 6. Therefore,
when a die is rolled getting a number
is decided by the face showing the number upward. As you know it could be any
number between 1 and 6 including them. Therefore,,
our sample space from a die
rolling is
P (3) = 

Empirical
Probability:
Probabilities can be computed for situations that do not use sample space. In
that case frequency distributions are used and the probability is called
Empirical probability.
Example: Imagine a class where,
Freshmen  4 
Sophomore  8 
Juniors  6 
Seniors  7 
Total  25 
P (E) = 

Example: To calculate the probability to pick a junior from above distribution, we may use the above formula
Frequency of juniors=6 and sum of frequency=25; therefore the probability to pick a junior from the class is
P (Junior) = 

Probability Rules: Probability should be given in a number between 0 and 1 or from 0% to 100%. Let's say probability of an event is P(E) then
If P (E) = 0 then the event E is not likely to happen at all and when P (E) = 1 then the event E will most likely to happen.
And if probability of an event to happen is P (E) then probability that event will not happen is 1P (E).
Examples for zero probability is a tossed up coin never falls to the ground or getting a 7 from a die and on the other hand, an example for the probability being =1, is an event of getting either head or tail from a coin toss.
A real world problem (1)
Shown above was a basket having
3 red balls, 11 green balls, and 7 blue balls that is total of (3 + 11 + 7) = 21
balls altogether. Now we do some exercise on some classical probability
problems.
Sorry about the poor illustration.
I know it's not up to the standards,
nevertheless, you have to excuse me since I am not an artist.
Answers: We assume that you place the ball back in to the basket after each pick.
P (E) = 

P(Red) = 

Now you may try to find the answers for questions 2 and 3 exactly the same way I solved the problem 1.
Drawing a card from a deck: As you know a deck of cards has 52 cards. There are four different kinds of suites namely Heart, Diamond, Club, and Spades. While Hearts and Diamonds cards are red in colors, Club and Spades are black. Each kind has 13 different cards. For an example Diamond has 13 different cards whose numbers are staring from A(ce) then 2, 3, 4, 5, 6, 7, 8, 9, 10, J(ack), Q(ueen) and K(ing). Likewise, Hearts, Clubs and Spades also have their own sets of 13 cards. That gives us total of 52 cards since 4 x 13 = 52. Therefore, out of 52 cards, we know there are 26 black cards and 26 red cards in a deck, and there should be four K's, four Q's ... and four 2's and so on. We also know that there are two black K's and two red K's. All the J's, K's and Q's are printed with a human face, so there are total of 12 cards have face on them of which, 6 are black and the other 6 are red. These are the few basic facts we should know about a deck of cards before trying to solve a probability question involving drawing a card from a deck.
A real world problem (2)
Sample Questions:
Imagine you are given a deck
of card. We assume that all the cards have equal chance to be picked and they are
all look the same when they are placed face down.
We will solve the problem 1 now. We have to find the probability of drawing a Diamond K out of 52 cards. Out of 52 cards there is only one Diamond King, therefore, Number of favorable outcome is 1, but Number of total possible outcomes =52, so the probability is
P(Diamond King)= 

Now problem 2 and 3 are very similar to problem 1 so I leave for you to answer.
We try to solve problem 4 now. We need to find the probability of picking a red card from the deck. Since we know that there are 26 red cards in a deck, so Number of favorable outcomes = 26 and Number of Total possible outcomes = 52 therefore the probability is
P (Red Card)= 

We have to give the answer in the smallest possible fraction. Therefore the answer is
P (Red Card)= 

Probability also can be given in percentage, like a weather man gives a forecast about a rain shower next day. He would usually say tomorrow there is a 30% chance of rain. Likewise we can also say in case of preceding example that the probability of a red card being drawn is 50%.
Law of large numbers: As we have learned already probability can't predict anything, I don't want any of the readers of this article to think that when coin is tossed every other time the result would be same. What probability suggest is that the chance of getting either head or tail is same. However, lets say a coin is tossed 1000 times, having a head or tail is close to being 500. If you increase the number of test by many fold having a head or tail for equal number of time becoming closer and closer.