📂Probability Theory

Independence of Events and Conditional Probability

Definition ¹

Suppose a probability space $(\Omega , \mathcal{F} , P)$ is given.

1. For $P(B)>0$, $\displaystyle P (A | B) = {{P(A \cap B)} \over {P(B)}}$ is called the conditional Probability of $A$ given $B$.
2. If $P(A | B) = P(A)$, that is, $P( A \cap B) = P(A) \cdot P(B)$, then $A, B$ are said to be independent of each other.

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• If you have not yet encountered measure theory, you may ignore the term probability space.

Explanation

Since the probability space is well defined, conditional probability and the independence of events can use the high school level definitions as they are. Because conditional probability and independence themselves are defined so intuitively well, this is only natural. The reason for bothering to mention this is to point out that ’nothing changes just because we introduced measure theory’, and to emphasize the contrast with independence and conditional expectation when discussing random variables.

As variations of conditional probability, we obtain the following two laws. These can be immediately applied to Bayes’ law.

Theorem

• [1] Multiplication law of probability: $$ P(A \cap B) = P(B) P(A | B) $$
• [2] Law of total probability: $$ P(C) = \sum_{i=1}^{k} P(C_{i}) P (C|C_{i}) $$