Introduction to Probability

There are lots of phenomena in nature, like tossing a coin or drawing cards from a card deck, whose outcomes cannot be predicted with certainty in advance, but the set of all the possible outcomes is known. These are what we call random phenomena or random experiments. Probability is concerned with such random phenomena or random experiments.
Probability is a way to measure the likelihood of certain outcomes of a particular activity that is typically random. 
Probability Notation
In general, the probability of an event is denoted by a capital P followed by the event put in paranethesis.
Example 1: If we wish to express “the probability it will snow next week,” we use the notation: P(snow next week).
Example 2: Lets go ahead and create a random variable that represents the outcome of a roll of a die. If we wished to find the probability that X = 2, we would simply express this as: P(X = 2).
Example 3: If we let A represent an event we wish to find the probability of, then P(A) would represent that probability.
P(win lottery) the probability that a person who has a lottery ticket will win that lottery
P(A) the probability that event A will occur
P(X = 2) the probability that random variable X equals 2
An outcome is the result of an experiment or random activity that involves uncertainty.
An event is any outcome or combination of outcomes. In other words, an event is a subset of the sample space. The probability of an impossible event is 0; the probability of a certain event is 1. Therefore, the range of possible probabilities for any given event is: 0≤P(A)≤1.
For any event A, P(A) must be a number between 0 and 1. ⇒  (0 \(\le\) P( A)  \(\le\) 1)

Sample Space

The sample space is the set of all the possible outcomes in an experiment. The sample space is denoted by a capital letter S. If you were to roll a single die, then S = {1, 2, 3, 4, 5, 6}, which represents the set of all possible outcomes. If you were to roll two dice simultaneously and look at the sum of the two dice, then S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. 
The probability of the Sample Space, S, is equal to one. ⇒ P(S) = 1 
Relative Frequency
Relative frequency refers to the proportion of times an event occurs. In other words, it is the number of times a particular outcome occurs divided by the total number of trials. In general, the probability of an event can be approximated by the relative frequency.  
Law of Large Numbers
The Law of Large Numbers is an important component of probability experiments. It states that as the number of repetitions or trials of an experiment increases, the relative frequency obtained in the experiment tends to become closer and closer to the theoretical probability, or true probability. Even though the short-term or immediately observed outcomes may vary widely, the Law of Large Numbers allows us to say that the long-term observed relative frequency will approach the theoretical probability.