**Probability is the branch of mathematics
concerned with some experiments called random experiments in which it is known
all the possible results, but there is no a certainty of what will be a specific
result. For example, random probability experiments are tossing a coin, tossing
a six-sided die, drawing a card from a standard deck of 52 plating cards.**

**The examples used above correspond to what
is called the mathematical or theoretical probability, that must be distinguished
from the statistical or empirical probability.**

**Some Probability Definitions.-**

__Sample Space.-__**Associated
to a random experiment, it is called Sample Space (S) to the set of all the
possible results for that experiment. **

**When tossing a coin, the sample space
is S={landing head, landing tail} or S={h, t} **

** When tossing a die, the sample space
is S={landing 1, landing 2, landing 3, landing 4, landing 5, landing 6} or
S={1, 2, 3, 4, 5, 6}.**

**When tossing two coins, the sample
space is S={(h,h), (h,t), (t,h), (t,t)}**

**When tossing three coins, the sample
space is S={(h,h,h), (h,h,t),(h,t,h),(h,t,t),(t,h,h),(t,h,t),(t,t,h),(t,t,t)}**

__Event.-__**An
event is any subset of the sample space. For example, in the sample space
S={1,2,3,4,5,6} when tossing a die, examples of events are:**

**1. To get a prime number, A={2,3,5}
2. To get a prime number and even, B={2}
3. To get a number equal or greater than 5,C={5,6}**

__Mutually Exclusive Events.-__**Two or more events are mutually exclusive
if not more than one of them can occur in a single trial, that is if and only
if its intersection is empty. For example, when tossing a die the events B={2}
and C={5,6} are mutually exclusive because their intersection is empty.**

__Complementary Events.-__**If AB
= and AB=S,
it is said that A and B are complementary events:
A ^{c}=B
and B^{c}=A **

__Mathematical Probability.-__**If in a random experiment all the results
have the same probability, that is, the occurrence of one result is equally
possible than the occurrence of any of the other results, then the probability
of an event A is the ratio:
P(A)=number of cases favorable to A/ total number of cases **

**From this definition, the probabilities
of the possible results of the experiment can be determined without putting
into effect the experiment. **

**From the definition is deduced that:
0 P(A)
1 The probability is a real number between 0 y 1, inclusive, or 0%
P(A) 100%
in percentage. P()
= 0 and P(S) =1**

__Statistical or Empirical Probability.-__**The relative frequency of the A result
of an experiment is the rate
F _{R}=number of times A occurs/ number of times the experiment is put into effect**

**If the experiment is repeated a great
number of times, The F _{R} value will approach to the probability
P of the event. For example, if I toss 100 times a coin, the number of times
I get head is nearly 50, that is F_{R} is close to 50%.**

**Probability Sets
Mathematics, Help Scope**