Friday, August 1, 2014

Unizor - Probability - Independent Events

It's natural to consider a concept of independent events as related to a concept of conditional probability. After all, "independence" of a random experiment from certain condition means that, no matter what that condition is, the outcomes from our experiment are the same and their chances to occur are also the same as if this condition never was imposed.

Let's call a random event A independent of a random event B if and only if the probability of A is the same as the conditional probability of A under condition of occurrence of an event B:
P(A) = P(A|B).

Mini Theorem 1
Independence is a symmetrical property of the events.
From independence of one event from another follows their mutual independence.
More precisely, if a random event A is independent of a random event B then a random event B is independent of a random event A.
Proof
By definition of conditional probability,
P(B|A) = P(A∩B) / P(A) and P(A|B) = P(A∩B) / P(B).
Let's resolve the second equation for P(A∩B) and substitute it into the first.
P(A∩B) = P(B) · P(A|B)
P(B|A) = P(B) · P(A|B) / P(A)
Now we can use the independence of a random event A of a random event B, which means that P(A|B)=P(A) and the right side of the previous equation equals to
P(B) · P(A) / P(A) = P(B).
Therefore,
P(B|A)=P(B),
which means that a random event B is independent of a random event A.
End of proof.

Using the above theorem, we can always replace the words mutually independent events with just independent events.

Mini Theorem 2
If random events A and B are independent then
P(A∩B) = P(A) · P(B)
Proof
By definition of conditional probability,
P(B|A) = P(A∩B) / P(A).
Since our events are independent,
P(B|A) = P(B).
Therefore,
P(B) = P(A∩B) / P(A) and
P(A∩B) = P(A) · P(B).
End of proof.

Mini Theorem 3
This is a converse theorem to a previous one.
If P(A∩B) = P(A) · P(B)
then random events A and B are independent.
Proof
By definition of conditional probability,
P(A|B) = P(A∩B) / P(B).
Since the probability of intersection of these events equals to a product of their respective probabilities,
P(A|B) = P(A) · P(B) / P(B).
Therefore,
P(A|B) = P(A),
which is a definition of independence.
End of proof.

Based on the mini-theorems 1, 2 and 3 above, we can equivalently define events A and B as independent, if they satisfy the following rule:
Probability of their intersection is equal to a product of their respective probabilities, that is
P(A∩B) = P(A) · P(B)