Question

If \(A\) and \(B\) are two events such that \[ P(A \cup B)=\frac{5}{6}, \qquad P(A \cap B)=\frac{1}{3}, \qquad P(B')=\frac{1}{2}, \] then the events \(A\) and \(B\) are:

• A. dependent
• B. mutually exclusive
• C. independent
• D. equally likely

Hint:

To test for independence, calculate the product \(P(A) \times P(B)\).
If it equals \(P(A \cap B)\), the events are independent.
If \(P(A \cap B) = 0\), the events are mutually exclusive.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Two events, \(A\) and \(B\), are statistically independent if the occurrence of one does not affect the probability of the occurrence of the other.

Step 2: Key Formula or Approach:
The mathematical condition for two events \(A\) and \(B\) to be independent is:
\[ P(A \cap B) = P(A) \times P(B) \]
We can find individual probabilities using the addition theorem:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Step 3: Detailed Explanation:
We are given:
\[ P(A \cup B) = \frac{5}{6} \]
\[ P(A \cap B) = \frac{1}{3} \]
\[ P(B') = \frac{1}{2} \]
First, find the probability of event \(B\) using the complement rule:
\[ P(B) = 1 - P(B') = 1 - \frac{1}{2} = \frac{1}{2} \]
Next, find the probability of event \(A\) using the addition theorem:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substitute our known values into the equation:
\[ \frac{5}{6} = P(A) + \frac{1}{2} - \frac{1}{3} \]
Simplify the fractional subtraction on the right-hand side:
\[ \frac{1}{2} - \frac{1}{3} = \frac{3 - 2}{6} = \frac{1}{6} \]
Substitute this back:
\[ \frac{5}{6} = P(A) + \frac{1}{6} \]
Solve for \(P(A)\):
\[ P(A) = \frac{5}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3} \]
Now, test the condition for independence by multiplying \(P(A)\) and \(P(B)\):
\[ P(A) \times P(B) = \frac{2}{3} \times \frac{1}{2} = \frac{2}{6} = \frac{1}{3} \]
Since our given joint probability is \(P(A \cap B) = \frac{1}{3}\), we have:
\[ P(A \cap B) = P(A) \times P(B) \]
Since this condition is satisfied, the events \(A\) and \(B\) are independent.

Step 4: Final Answer:
Therefore, the correct option is (C).