Probabilities

1. Definitions

Probability finds its origin in gambling. Monte Carlo calculation is an 
example. Probabily theory is a set of rules and methods that we use to assess 
the chances to get an event, and take, thereafter, the best decision 
regarding the established circumstances.

Searching for a value for a probobability that an event occurs, requires 
an action for which we do not know the outcome in advance, that is an aleatory
experiment. Tossing a fair coin leads to two possible outcomes: head or tail. 
This action is an aleatory experiment. Throwing a dice is another one. In 
other words, the action is a random trial or random experiment.

Probability remains the mean to measure or quatify "how likely" an event, 
related to a random experiment, will happen. The value of a probability 
is a number between 0 and 1 inclusive. An event that is certain to 
occur has a probability equal to 1. An event that cannot occur has a 
probability equal to 0. 


A sample space or universal sample space is the set of all possible 
outcomes. It's also referred to parent population We will note it as S 
and |S| the number of elements that it contains. 
A sprobability space is called an event. It is a subset of the set 
S associated to certain random trial. In other words, an event is a subset of 
S with certain conditions, that is for a very special case.

2. Examples

1. Tossing a coin one time will have S = {Head, Tail} or {H,T}

2. Tossing a coin two times will have S = {(H,H),(H,T),(T,H),(T,T)}

3. Rolling a die will  have S = {1,2,3,4,5,6} 

4. Rolling two dices will have 
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)}

5. Three coins are tossed. If we define the event E as 
the set of possible outcomes with at least one head, we 
will have: E = {(H,H),(HT),(TH)}

6. Two six-sided dice are thrown. If we define the event E as the set 
of possible outcomes where the sum of the numbers is equal to five, 
we will have:
E = {(1,4),(2,3),(3,2),(4,1)} 

3.Equally likelihood

Let's consider the above example of rolling two six-sided dice with 
the condition to "obtain the sum of the outcome numerbs equal to five".

If we write S as the set of all possible outcomes as 
S = {2,3,4,5,6,7,8,9,10,11,12}; then the event E = {5}. 
This event can occur with four (1,4), (4,1), (2,3), or (3,2). 

If the condition to "obtain the sum of the outcome numerbs equal to twelve", 
we will have just the two outcomes: (6,6) or (6,6).

We conclude that the two events or the possible outcomes are not 
equally likely. 
But if S = {(1,1),(1,2),(1,3), ... (2,1),(2,2), ...(6,6)} (|S| = 36)
then E = {(1,4),(4,1),(2,3),(3,2)}. Here all of the outcomes are 
equally likely.

This definition is crucial to define the probability of an event.