1. Key Concepts and Formulas

• Conditional Probability: The probability of an event $E$ occurring given that another event $F$ has already occurred is denoted by $P(E|F)$. It quantifies how the occurrence of one event influences the likelihood of another.
• Formula for Conditional Probability: The formal definition is: $P(E|F) = \frac{P(E \cap F)}{P(F)}$ where $P(E \cap F)$ is the probability of both events $E$ and $F$ occurring (their intersection), and $P(F)$ is the probability of event $F$ occurring. It's essential that $P(F) > 0$.
• Intersection of Events: The probability of two events $A$ and $B$ both occurring is denoted by $P(A \cap B)$ (or $P(B \cap A)$), and it can be expressed in terms of conditional probabilities as: $P(A \cap B) = P(A|B) \times P(B)$ $P(A \cap B) = P(B|A) \times P(A)$

2. Step-by-Step Solution

We are given the following probabilities:

• $P(A) = \frac{1}{4}$
• $P(A|B) = \frac{1}{2}$
• $P(B|A) = \frac{2}{3}$

Our goal is to find $P(B)$.

Step 1: Strategize - Identify the common link.

• Why this step? We have information about conditional probabilities involving both $A$ and $B$, along with $P(A)$. We need to find $P(B)$. The term $P(A \cap B)$ (the probability of both events occurring) is present in both conditional probability formulas. This makes $P(A \cap B)$ a crucial intermediate value that can connect the given information to the desired $P(B)$.
• Our plan:
  1. Use the formula for $P(B|A)$ and the given $P(A)$ to calculate $P(A \cap B)$.
  2. Once $P(A \cap B)$ is known, use the formula for $P(A|B)$ and the given $P(A|B)$ to calculate $P(B)$.

Step 2: Calculate the probability of the intersection, $P(A \cap B)$.

• Why this step? We have direct values for $P(B|A)$ and $P(A)$. The formula $P(B|A) = \frac{P(A \cap B)}{P(A)}$ allows us to directly solve for $P(A \cap B)$, which is a necessary intermediate step.

• Applying the formula: Start with the definition of $P(B|A)$: $P(B|A) = \frac{P(A \cap B)}{P(A)}$ Substitute the given values: $P(B|A) = \frac{2}{3}$ and $P(A) = \frac{1}{4}$: $\frac{2}{3} = \frac{P(A \cap B)}{\frac{1}{4}}$

• Solving for $P(A \cap B)$: To find $P(A \cap B)$, multiply both sides of the equation by $P(A) = \frac{1}{4}$: $P(A \cap B) = \frac{2}{3} \times \frac{1}{4}$ $P(A \cap B) = \frac{2 \times 1}{3 \times 4}$ $P(A \cap B) = \frac{2}{12}$ Simplify the fraction: $P(A \cap B) = \frac{1}{12}$ So, the probability that both event $A$ and event $B$ occur is $\frac{1}{12}$.

Step 3: Calculate the probability of event $B$, $P(B)$.

• Why this step? Now that we have calculated $P(A \cap B)$, we can use the other conditional probability given, $P(A|B)$, to find $P(B)$. The formula $P(A|B) = \frac{P(A \cap B)}{P(B)}$ contains $P(B)$ as the only unknown, allowing us to solve for it.

• Applying the formula: Start with the definition of $P(A|B)$: $P(A|B) = \frac{P(A \cap B)}{P(B)}$ Substitute the given value $P(A|B) = \frac{1}{2}$ and our calculated value $P(A \cap B) = \frac{1}{12}$: $\frac{1}{2} = \frac{\frac{1}{12}}{P(B)}$

• Solving for $P(B)$: To isolate $P(B)$, we can rearrange the equation. Multiply both sides by $P(B)$ and then divide by $\frac{1}{2}$: $P(B) = \frac{\frac{1}{12}}{\frac{1}{2}}$ Recall that dividing by a fraction is equivalent to multiplying by its reciprocal: $P(B) = \frac{1}{12} \times \frac{2}{1}$ $P(B) = \frac{2}{12}$ Simplify the fraction: $P(B) = \frac{1}{6}$

Thus, the probability of event $B$ is $\frac{1}{6}$.

3. Common Mistakes & Tips

• Master the Formulas: Always start by writing down the definitions of conditional probability and the intersection of events. This ensures you're using the correct relationships.
• Trace the Connections: Many probability problems require finding an intermediate value (like $P(A \cap B)$ here) to connect the given information to the desired unknown. Identifying this "bridge" early is key.
• Algebraic Precision: Be meticulous with algebraic manipulations, especially when dealing with fractions. Errors in multiplication, division, or simplification are common pitfalls. Remember that $\frac{a/b}{c/d} = \frac{a}{b} \times \frac{d}{c}$.
• Distinguish Conditional Probabilities: $P(A|B)$ is generally not equal to $P(B|A)$. Understanding which event is the condition and which is the event whose probability is being assessed is crucial.
• Check for Validity: A probability must always be between 0 and 1, inclusive. If your answer falls outside this range, recheck your calculations.

4. Summary

To determine $P(B)$, we utilized the fundamental definitions of conditional probability. First, we used the given $P(B|A)$ and $P(A)$ to calculate the probability of the intersection of events $A$ and $B$, finding $P(A \cap B) = \frac{1}{12}$. Subsequently, we employed this calculated value along with the given $P(A|B)$ to solve for $P(B)$, which we found to be $\frac{1}{6}$.

5. Final Answer

The final answer is $\boxed{\text{1/6}}$, which corresponds to option (A).