# Conditional Probability

## Key Questions

Using the ex-post probabilities, it tries to calculate ex-ante probability.

#### Explanation:

Suppose there are 3 production units A, B and C. They are used to produce Good-X. Again, the probabilities of a randomly selected item from A, B and C are known.

If we mixed up all their output. Now, if we select one item, we do not know from which production unit, it has come from.

O.K., Suppose the selected Item is defective, we can find out the probability about which production unit has produced it.

• General Multiplication Rule of Probability is related to a probability of a combined occurrence of any two events $A$ and $B$ (denoted as $A \cdot B$) expressed in term of their individual (denoted as $P \left(A\right)$ and $P \left(B\right)$ correspondingly) and conditional probabilities (probability of occurrence of one event under condition of occurrence of another, denoted as $P \left(A | B\right)$ and $P \left(B | A\right)$ correspondingly):

$P \left(A \cdot B\right) = P \left(A\right) \cdot P \left(B | A\right) = P \left(B\right) \cdot P \left(A | B\right)$

Special Multiplication Rule is related to a probability of a combined occurrence of two independent events (that is, the probability of one is not dependent on the probability of another or, in other words, conditional probability of one under condition of occurrence of another equals to its unconditional probability). This necessitates $P \left(A | B\right) = P \left(A\right)$ and $P \left(B | A\right) = P \left(B\right)$, and the multiplication rule looks like this:

$P \left(A \cdot B\right) = P \left(A\right) \cdot P \left(B\right)$

Informative lectures and solutions to many problems of the Theory of Probabilities for beginners can be found in the corresponding chapter on the Web site Unizor (free online course of advanced mathematics for teenagers). There you can find definitions and properties of main objects the Theory of Probabilities deals with - sample space, events, measures of probability, independence etc.