Understanding Event Independence in Probability

This solution explains how to determine if two events, A and B, are independent, dependent, or mutually exclusive based on their given probabilities. We will use the provided values for \( \rm P(A\cup B) \), \( \rm P(A\cap B) \), and \( \rm P(\bar{B}) \) to perform the necessary calculations.

Key Probability Concepts

• Events A and B are independent if the occurrence of one event does not change the probability of the other event occurring. The condition for independence is \( \rm P(A \cap B) = P(A) \cdot P(B) \).
• Events A and B are mutually exclusive if they cannot happen at the same time. This means their intersection is empty, so \( \rm P(A \cap B) = 0 \).
• Dependent events are events where the outcome of one event influences the outcome of the other.
• The probability of the complement of event B, denoted as \( \rm P(\bar{B}) \), is related to the probability of event B, \( \rm P(B) \), by the formula: \( \rm P(B) = 1 - P(\bar{B}) \).
• The addition rule for any two events A and B states: \( \rm P(A \cup B) = P(A) + P(B) - P(A \cap B) \).

Given Information

Step-by-Step Solution

1. Calculate the probability of event B, \( \rm P(B) \):

   We use the relationship between the probability of an event and its complement.

   Formula: \( \rm P(B) = 1 - P(\bar{B}) \).

   Substituting the given value: \( \rm P(B) = 1 - \dfrac{1}{2} = \dfrac{1}{2} \).

2. Calculate the probability of event A, \( \rm P(A) \):

   We use the addition rule for probabilities.

   Formula: \( \rm P(A \cup B) = P(A) + P(B) - P(A \cap B) \).

   Rearranging the formula to solve for \( \rm P(A) \): \( \rm P(A) = P(A \cup B) - P(B) + P(A \cap B) \).

   Substituting the known values: \( \rm P(A) = \dfrac{5}{6} - \dfrac{1}{2} + \dfrac{1}{3} \).

   To perform the calculation, find a common denominator, which is 6:

   \( \rm P(A) = \dfrac{5}{6} - \dfrac{3}{6} + \dfrac{2}{6} = \dfrac{5 - 3 + 2}{6} = \dfrac{4}{6} \).

   Simplifying the fraction: \( \rm P(A) = \dfrac{2}{3} \).

3. Check if events A and B are independent:

   The condition for independence is \( \rm P(A \cap B) = P(A) \cdot P(B) \).

   First, calculate the product \( \rm P(A) \cdot P(B) \):

   \( \rm P(A) \cdot P(B) = \dfrac{2}{3} \cdot \dfrac{1}{2} = \dfrac{2}{6} = \dfrac{1}{3} \).

   Now, compare this result with the given \( \rm P(A \cap B) \).

   Given: \( \rm P(A \cap B) = \dfrac{1}{3} \).

   Since \( \rm P(A \cap B) = \dfrac{1}{3} \) is equal to \( \rm P(A) \cdot P(B) = \dfrac{1}{3} \), the condition for independence is met.

4. Check if events A and B are mutually exclusive:

   Events are mutually exclusive if \( \rm P(A \cap B) = 0 \).

   We are given \( \rm P(A \cap B) = \dfrac{1}{3} \).

   Since \( \rm P(A \cap B) \neq 0 \), the events are not mutually exclusive.

Conclusion

The calculations show that \( \rm P(A \cap B) \) is equal to the product \( \rm P(A) \cdot P(B) \). This satisfies the definition of independent events. Therefore, events A and B are independent.