Probability properties

1. Definition of a probability

If S is a sample space associated to a random experiment. For 
each event E, the probability for this event p(E) is a number 
that must satisfy the following conditions:

1. 0 ≤ P(E) ≤ 1
2. P(S) = 1
3. P(E1 ∪ E2) = P(E1) + P(E2) + P(E1 ∩ E2)

2. Properties

1. An event E1 implies an event E2 if each time E1 occurs, 
then E2 ocuurs as well. 
If an event E1 implies an event E2, then the probability 
of E1 is lesser that the probability of E2:
If E1 ⊆ E2 then P(E1) ≤ P(E2)


2. The sum of the probabilities of two contrary events 
is equal to 1. If E occurs, then E will not occur:
P(E + E)=1


3. Events are mutually exclusive or incompatible 
if both cannot be achieved simultanuously. In terms of sets, they 
are disjoint events.
If E1, E2, ... En are mutually exlusive: E1 ∩ E2 ∩ ... ∩ 
En = ∅  then:
P(E1 ∪ E2 ... ∪ En) = P(E1) + P(E2) + ... + P(En)


4. The probabilty of an event is equal to the sum of the 
probabilities of each element that compose this event:
If E = {X1, X2, ... Xn}, then 
P(E) = P({X1}) + P({X2}) + ... + P({Xn})


5. If E1, E2, ... En are events associated to a random 
experiment, then:
P(E1 ∪ E2 ... ∪ En) = ∑ P(Ei) - ∑ P(Ei ∩ Ej≠i)

3. Conditional probability

Let's consider again the random experiment of rolling two six-sided 
dice. We know that:
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)}

For the event E "have the sum of two outcome numbers equal to 5", we 
have E = {(1,4),(2,3),(3,2),(4,1)},
Now for another event Eex "the outcome number one (1) is exluded", 
we have 
Eex = {(2,2),(2,3),(2,4),(2,5),(2,6),(3,2),(3,3),(3,4),(3,5),(3,6),(4,2),(4,3),
(4,4),(4,5),(4,6),(5,2),(5,3),(5,4),(5,5),(5,6),(6,2),(6,3),(6,4),(6,5),(6,6)}
|Eex| = 25

What is the probability of the event E according to the event Eex? 
It is not anymore 4/36 = 1/9.
The event Eex imposes a condition for the event E. The probability of the 
event E becomes conditional regarding the event Eex.

The conditional probabily of an event E1 
according to the event E2 is P(E1|E2) = P(E1 ∩ E2)/P(E2)


For our example:
E ∩ Eex = {(2,3),(3,2)}
P(E|Eex) = P(E ∩ Eex)/P(Eex) = (2/36)/(25/36) = 2/25

Two events E1 and E2 are independents if their occurences do 
not influence each other; that is one is not conditional to another. 
This can be written as: P(E1|E2) = P(E1)
Using the above formula, it follows:
P(E1|E2) = P(E1) = P(E1 ∩ E2)/P(E2) → P(E1) x P(E2) = P(E1 ∩ E2).

Two events E1 and E2 are independents if: 
P(E1 ∩ E2) = P(E1) x P(E2)

Independent does not mean mutually exlusive.

4. Bayes' Formula

E1, E2, ... En is a partition of a set S if they are 
mutually exclusive and their union gives S, that is:
∩ Ei = ∅
∪ Ei = S

Bayes' Formula: 
if Ei (E1, E2, ... En) is a partition of the 
sample space S associated to a random experiment, 
and E an event, then: for any number "r" between 
1 and k, we have:
P(Er|E) = P(Er) x P(E|Er)/∑ P(Ei) x P(E|Ei)	(r: i → k)