Question

When A and B are dependent, the probability of occurence of B is affected by the occurence of A. We must use it in its original form. •Thus, if A and B are dependent: p(A and B) = p(A) p(B|A) Sampling without replacement provides a good illustration for dependent events.

Answer 1

Dependent and independent events in probability have distinct characteristics: dependent events influence each other's probabilities while independent events do not.

Step-by-step explanation:

Two events are considered dependent if the occurrence of one affects the probability of the occurrence of the other.

An example of dependent events is sampling without replacement, where the sample space is reduced after each selection, changing the probabilities of subsequent selections.

In contrast, independent events do not influence each other's probabilities, as in the case of sampling with replacement.

The mathematical formulation for the probability of both A and B occurring in dependent events is P(A AND B) = P(A) \(\times\) P(B|A).

For independent events, it simplifies to P(A AND B) = P(A) \(\times\) P(B). Therefore, to determine whether two events are independent or dependent, you can check if P(A AND B) equals P(A) \(\times\) P(B).

Understanding the relationship between events is critical in probability theory, as it affects the method used to calculate the combined probability of multiple occurrences.

The conditional probability, contingency table, and the product rule for independent events are examples of concepts used in such calculations.