Determining Independence of Events A and B Using Probability Theory
In probability theory, determining whether two events are independent or dependent is a fundamental concept. This article explores the relationship between events A and B given their individual probabilities and the probability of their union. Through the application of the third axiom of probability, we will determine if A and B are independent or dependent.
Given Information
Let us consider events A and B such that:
PA 1/2 PB 1/3 P(A ∪ B) 2/3Our objective is to determine whether A and B are independent or dependent. Two events are independent if and only if the following condition holds true:
(text{P}(A cap B) text{P}(A) times text{P}(B))
Step-by-Step Analysis
Using the Third Axiom of Probability
The third axiom of probability states that the probability of the union of two events can be found using the following formula:
(text{P}(A cup B) text{P}(A) text{P}(B) - text{P}(A cap B))
Substituting the given values:
(frac{2}{3} frac{1}{2} frac{1}{3} - text{P}(A cap B))
(frac{2}{3} frac{5}{6} - text{P}(A cap B))
(text{P}(A cap B) frac{5}{6} - frac{2}{3} frac{1}{6})
Checking for Independence
To verify if A and B are independent, we must check if the following condition holds:
(text{P}(A cap B) text{P}(A) times text{P}(B))
Calculating the Right Side:
(frac{1}{6} frac{1}{2} times frac{1}{3} frac{1}{6})
Since the values on both sides are equal, we can conclude that A and B are independent.
Summary of Findings
From the calculations, it is evident that A and B are independent because:
(text{P}(A cap B) frac{1}{6}) (text{P}(A) times text{P}(B) frac{1}{2} times frac{1}{3} frac{1}{6})Furthermore, from the given information, we can determine that A and B are not mutually exclusive because the probability of their union is not simply the summation of their individual probabilities but rather the sum minus the probability of their intersection.
By applying the rules of probability theory, we have confirmed that A and B are indeed independent events.