Conditional Probability Definition

The conditional probability of an event B in related to an event A is the probability that event B happens given that event A has already occurred. The notation for conditional probability is P(B|A), read as the probability of B given A. this formula comes from the multiplication principle and bit of algebra.

P (B | A) = P (A and B) / P (A)

The given event A has occurred; we should a reduced sample space. Instead of the entire sample space S, now we have a sample space of A since we know A has occurred.  So the old thing is about being the number in the event divided by the number in the sample space still applies.

It is the number in A and B (should be in A since A has happened) divided by the number in A. By dividing on right hand side by the number which is present in the sample space S, and then we have the probability of A and B divided by the probability of A. Another important method of conditional probabilities is given by Bayes's formula.

Conditioning on a random variables:

Conditional probability of an event a discrete random variable. Such a conditional probability is a random variable in its belongs right.

Let X is a random variable it can be equal to 0 or to 1. As above, the conditional probability of any event A given the event X = 0, and also of the conditional probability of A given the event X = 1. The former is placed by P (A|X = 0) and the latter P(A|X = 1). Define a new random variable Y, which value is P (A|X = 0) if X = 0 and P(A|X = 1) if X = 1. That is,

Y = P(A | X = 0 ) if X = 0

P(A | X = 1 ) if X = 1

variables in the conditional probability:

Random variable Y is saying that to be the conditional probability of the event A given the discrete random variable X:

Y = P ( A | X )

Conditioning, the probability of A given initial information I, P (A|I), is known as the prior probability and updating conditional probability of A, given I and the outcome of the event B, is known as the posterior probability, P (A|B,I).

Bayes's formula.

It is based on the expression P(B) = P(B|A)P(A) + P(B|Ac)P(Ac),

P(A|B) = P(B|A)P(A) / P(B|A)P(A) + P(B|Ac) P(Ac)