Key Concepts and Formulas

To solve this problem, we'll utilize several fundamental concepts from probability theory:

• Complement Rule: The probability of an event $E$ not occurring, denoted as $P(\overline E)$, is related to the probability of $E$ occurring, $P(E)$, by the formula: $P(E) = 1 - P(\overline E)$
• Addition Rule for Probability: For any two events $A$ and $B$, the probability that at least one of them occurs ($A$ or $B$) is given by: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
• Independent Events: Two events $A$ and $B$ are considered independent if the occurrence of one does not affect the probability of the other. Mathematically, this condition is satisfied if: $P(A \cap B) = P(A) \cdot P(B)$
• Equally Likely Events: Two events $A$ and $B$ are equally likely if they have the same probability of occurrence: $P(A) = P(B)$
• Mutually Exclusive Events: Two events $A$ and $B$ are mutually exclusive (or disjoint) if they cannot occur at the same time. This means their intersection is an empty set, and consequently: $P(A \cap B) = 0$

Step-by-Step Solution

Our goal is to determine if events $A$ and $B$ are independent, equally likely, or mutually exclusive based on the given probabilities. We'll systematically calculate $P(A)$, $P(B)$, and $P(A \cup B)$ first.

Step 1: Calculate $P(A)$ and $P(A \cup B)$ using the Complement Rule.

• What we are doing: We are given probabilities of complements, $P(\overline A)$ and $P(\overline{A \cup B})$. To use the Addition Rule and check for independence, we need the probabilities of the events themselves, $P(A)$ and $P(A \cup B)$.

• Why this step is important: The complement rule is a direct way to convert probabilities of "not occurring" into probabilities of "occurring."

  a. Determine $P(A)$: We are given $P\left( {\overline A } \right) = {1 \over 4}$. Using the complement rule, $P(A) = 1 - P(\overline A)$: $P(A) = 1 - {1 \over 4} = {3 \over 4}$ So, the probability of event $A$ occurring is $\frac{3}{4}$.

  b. Determine $P(A \cup B)$: We are given $P\left( {\overline {A \cup B} } \right) = {1 \over 6}$. Using the complement rule, $P(A \cup B) = 1 - P(\overline{A \cup B})$: $P(A \cup B) = 1 - {1 \over 6} = {5 \over 6}$ So, the probability of at least one of events $A$ or $B$ occurring is $\frac{5}{6}$.

Step 2: Calculate $P(B)$ using the Addition Rule of Probability.

• What we are doing: We have $P(A)$, $P(A \cup B)$, and we are given $P(A \cap B)$. The Addition Rule is the formula that connects these three probabilities with $P(B)$.

• Why this step is important: To check if events are equally likely or independent, we need the individual probability $P(B)$.

  We have the following values:

  • $P(A \cup B) = {5 \over 6}$ (from Step 1)
  • $P(A) = {3 \over 4}$ (from Step 1)
  • $P(A \cap B) = {1 \over 4}$ (given in the question)

  Substitute these into the Addition Rule, $P(A \cup B) = P(A) + P(B) - P(A \cap B)$: ${5 \over 6} = {3 \over 4} + P(B) - {1 \over 4}$ Combine the fractions on the right side: ${5 \over 6} = \left( {3 \over 4} - {1 \over 4} \right) + P(B)$ ${5 \over 6} = {2 \over 4} + P(B)$ ${5 \over 6} = {1 \over 2} + P(B)$ Now, solve for $P(B)$: $P(B) = {5 \over 6} - {1 \over 2}$ To subtract, find a common denominator (which is 6): $P(B) = {5 \over 6} - {3 \over 6}$ $P(B) = {2 \over 6} = {1 \over 3}$ So, the probability of event $B$ occurring is $\frac{1}{3}$.

Step 3: Check for Independence of Events A and B.

• What we are doing: We will apply the definition of independent events, which states that $P(A \cap B)$ must be equal to $P(A) \cdot P(B)$.

• Why this step is important: This is the direct test to determine if the events are independent.

  We have:

  • $P(A \cap B) = {1 \over 4}$ (given)
  • $P(A) = {3 \over 4}$ (calculated in Step 1)
  • $P(B) = {1 \over 3}$ (calculated in Step 2)

  Now, let's calculate the product $P(A) \cdot P(B)$: $P(A) \cdot P(B) = {3 \over 4} \cdot {1 \over 3} = {3 \over {12}} = {1 \over 4}$ Compare this product with the given $P(A \cap B)$: $P(A \cap B) = {1 \over 4}$ Since $P(A \cap B) = P(A) \cdot P(B)$, the events $A$ and $B$ are independent.

Step 4: Check if Events A and B are Equally Likely.

• What we are doing: We will compare the individual probabilities of events $A$ and $B$ to see if they are equal.

• Why this step is important: This is the direct test to determine if the events are equally likely.

  We have:

  • $P(A) = {3 \over 4}$
  • $P(B) = {1 \over 3}$

  Comparing these probabilities: ${3 \over 4} \neq {1 \over 3}$ Since $P(A) \neq P(B)$, the events $A$ and $B$ are not equally likely.

Step 5: Check if Events A and B are Mutually Exclusive (Optional but useful).

• What we are doing: We will check if $P(A \cap B) = 0$.

• Why this step is important: Although the options primarily focus on independence and equally likely, understanding mutual exclusivity helps to differentiate it from independence, a common point of confusion.

  We are given $P(A \cap B) = {1 \over 4}$. Since $P(A \cap B) = {1 \over 4} \neq 0$, the events $A$ and $B$ are not mutually exclusive. This is consistent with them being independent and having non-zero probabilities, as independent events with non-zero probabilities cannot be mutually exclusive.

Common Mistakes & Tips

• Confusing Independence with Mutually Exclusive: These are distinct concepts. Independent events can occur together (their intersection probability is the product of individual probabilities). Mutually exclusive events cannot occur together (their intersection probability is zero). If $P(A) > 0$ and $P(B) > 0$, independent events cannot be mutually exclusive.
• Fraction Arithmetic Errors: Be meticulous with addition, subtraction, and multiplication of fractions. A small calculation error can lead to a completely different conclusion about the events. Always find a common denominator for addition/subtraction.
• Understanding Notation: Ensure you correctly interpret $P(\overline A)$, $P(A \cup B)$, and $P(A \cap B)$ to apply the correct formulas. For example, $P(\overline{A \cup B})$ means neither A nor B occurs, which is equivalent to $P(\overline A \cap \overline B)$.

Summary

We systematically calculated the individual probabilities $P(A) = \frac{3}{4}$ and $P(B) = \frac{1}{3}$ by first using the complement rule to find $P(A)$ and $P(A \cup B)$, and then applying the Addition Rule of Probability to find $P(B)$. With these values and the given $P(A \cap B) = \frac{1}{4}$, we tested for independence by checking if $P(A \cap B) = P(A) \cdot P(B)$ and found that they are indeed independent. We then checked if they are equally likely by comparing $P(A)$ and $P(B)$ and found that they are not. Therefore, events $A$ and $B$ are independent but not equally likely.

The final answer is $\boxed{A}$ which corresponds to option (A).