Key Concepts and Formulas

• Complement Rule: For any event $A$, the probability of $A$ occurring plus the probability of $A$ not occurring (its complement, $\overline A$) is equal to 1. Mathematically, $P(A) + P(\overline A) = 1$. This implies $P(A) = 1 - P(\overline A)$.
• Inclusion-Exclusion Principle for Two Events: This principle relates the probabilities of the union, intersection, and individual events $A$ and $B$. It states that $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. This formula accounts for outcomes that are common to both events, preventing them from being counted twice.
• Probability of "B only" (B occurs, A does not): The event $\overline A \cap B$ represents the outcomes where event $B$ happens, but event $A$ does not. This probability can be calculated as $P(\overline A \cap B) = P(B) - P(A \cap B)$. Visually, in a Venn diagram, this is the part of circle $B$ that does not overlap with circle $A$.

Step-by-Step Solution

Our goal is to find $P(\overline A \cap B)$. We will achieve this by first finding $P(A)$ and $P(B)$ using the given probabilities and the fundamental rules.

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Step 1: Calculate $P(A)$ using the Complement Rule.

• What we are doing: We are determining the probability of event $A$ occurring.
• Why this step is taken: We are given $P(\overline A)$, the probability that event $A$ does not occur. The Complement Rule provides a direct way to find $P(A)$ from $P(\overline A)$, and $P(A)$ is typically needed in other probability formulas, such as the Inclusion-Exclusion Principle.
• Calculation: Using the Complement Rule: $P(A) = 1 - P(\overline A)$ Substitute the given value $P(\overline A) = \frac{2}{3}$: $P(A) = 1 - \frac{2}{3}$ $P(A) = \frac{3}{3} - \frac{2}{3}$ $P(A) = \frac{1}{3}$ So, the probability of event $A$ is $\frac{1}{3}$.

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Step 2: Calculate $P(B)$ using the Inclusion-Exclusion Principle.

• What we are doing: We are finding the probability of event $B$ occurring.
• Why this step is taken: The problem asks for $P(\overline A \cap B)$, which can be expressed as $P(B) - P(A \cap B)$. We already have $P(A \cap B)$ (given), but we need $P(B)$. The Inclusion-Exclusion Principle connects $P(A \cup B)$, $P(A)$, $P(B)$, and $P(A \cap B)$. Since we now know $P(A \cup B)$, $P(A)$ (from Step 1), and $P(A \cap B)$, we can use this principle to solve for $P(B)$.
• Calculation: Using the Inclusion-Exclusion Principle: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ Substitute the known values:
  • $P(A \cup B) = \frac{3}{4}$ (given)
  • $P(A) = \frac{1}{3}$ (from Step 1)
  • $P(A \cap B) = \frac{1}{4}$ (given) $\frac{3}{4} = \frac{1}{3} + P(B) - \frac{1}{4}$ Now, rearrange the equation to solve for $P(B)$: $P(B) = \frac{3}{4} - \frac{1}{3} + \frac{1}{4}$ Group the fractions with common denominators: $P(B) = \left(\frac{3}{4} + \frac{1}{4}\right) - \frac{1}{3}$ $P(B) = 1 - \frac{1}{3}$ $P(B) = \frac{2}{3}$ So, the probability of event $B$ is $\frac{2}{3}$.

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Step 3: Calculate $P(\overline A \cap B)$ using the "B only" formula.

• What we are doing: We are determining the probability that event $B$ occurs, but event $A$ does not.
• Why this step is taken: This is the final step, directly calculating the probability requested in the problem. We have found $P(B)$ in Step 2, and $P(A \cap B)$ was given. These are the two components needed for this specific formula.
• Calculation: The probability of $B$ occurring and $A$ not occurring is given by: $P(\overline A \cap B) = P(B) - P(A \cap B)$ Substitute the values we have:
  • $P(B) = \frac{2}{3}$ (from Step 2)
  • $P(A \cap B) = \frac{1}{4}$ (given) $P(\overline A \cap B) = \frac{2}{3} - \frac{1}{4}$ To subtract these fractions, find a common denominator, which is 12: $P(\overline A \cap B) = \frac{2 \times 4}{3 \times 4} - \frac{1 \times 3}{4 \times 3}$ $P(\overline A \cap B) = \frac{8}{12} - \frac{3}{12}$ $P(\overline A \cap B) = \frac{8 - 3}{12}$ $P(\overline A \cap B) = \frac{5}{12}$

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Common Mistakes & Tips

• Venn Diagram Visualization: Always consider sketching a Venn diagram. It helps in understanding the relationship between events like $A$, $B$, $A \cap B$, $A \cup B$, and $\overline A \cap B$. For instance, $P(\overline A \cap B)$ is the part of $B$ that does not overlap with $A$.
• Notation Clarity: Be precise with the meaning of symbols: $\cup$ for "OR" (union), $\cap$ for "AND" (intersection), and $\overline{\phantom{A}}$ for "NOT" (complement). $\overline A \cap B$ specifically means "B occurs AND A does NOT occur."
• Fraction Arithmetic: Double-check all calculations involving fractions, especially when finding common denominators for addition and subtraction. A small arithmetic error can lead to a completely wrong answer.

Summary

This problem demonstrates a systematic approach to solving probability questions involving multiple events. We began by using the Complement Rule to find $P(A)$ from the given $P(\overline A)$. Subsequently, we employed the Inclusion-Exclusion Principle, along with the given $P(A \cup B)$ and $P(A \cap B)$, to determine $P(B)$. Finally, by understanding that $P(\overline A \cap B)$ represents the probability of event $B$ occurring exclusively (without $A$), we subtracted $P(A \cap B)$ from $P(B)$ to arrive at the desired probability. This methodical application of fundamental rules ensures accuracy and clarity in the solution.

The final answer is $\boxed{\text{5/12}}$, which corresponds to option (A).