3 - Conditional Probability

Prob of event occurring given another event has occurred, proba of intersection of 2 events, independence with intersections, chain rule for disjoint distribution, complement within a conditional

Conditional Probability

$P\left(A\left|B\right|\right)=\frac{P\left(A\cap B\right)}{P\left(B\right)}$ is the formula for conditional probability, which calculates the probability of event A occurring given that event B has occurred, provided that P(B) is greater than zero.

Probability of the Intersection of 2 events

$P\left(A\cap B\right)=P\left(A\left|B\right|\right)\cdot P\left(B\right)$ is the formula used to express the probability of both events A and B occurring simultaneously, showing that it can be calculated by the product of the conditional probability of A given B and the probability of B.

Independence with Intersections

Two events A and B are independent if and only if the probability of their intersection equals the product of their probabilities, stated as $P(A\cap B)=P(A)\cdot P(B)$ .

Chain Rule for Disjoint Distribution

$P\left(X_1,X_2,\ldots,X_{N}\right)=P\left(X_1\right)\cdot P\left(X_2\left|X_1\right|\right)\cdot P\left(X_3\left|X_1\cdot X_2\right|\right)\cdot\ldots\cdot P\left(X_{N}\left|X_1,\ldots,X_{N-1}\right|\right)$ provides a way to calculate the joint probability of multiple events by multiplying the probability of the first event by the conditional probabilities of subsequent events given all previous events.

Complement within Conditional Probabilities

The complement of an event A, denoted as $A^c$, represents the scenario where A does not occur. In the context of conditional probability, the probability of the complement can be expressed as $P(A^c|B) = 1 - P(A|B)$, indicating that the sum of the probabilities of an event and its complement equals one.

True or False:

$$P\left(A\left|B\right.\right)\ne P\left(B\left|A\right.\right)$$

True

Normalizing Concept

True or False:

All probabilities are conditional

True