Key Concepts and Formulas

• Conditional Probability: The probability of an event $A$ occurring given that another event $B$ has already occurred is denoted as $P(A|B)$. It is calculated as: $P(A|B) = \frac{P(A \cap B)}{P(B)}$ In problems involving counts, this can be more intuitively expressed as: $P(A|B) = \frac{\text{Number of outcomes where A and B both occur}}{\text{Number of outcomes where B occurs}}$
• Reduced Sample Space: When a condition is given (e.g., "chosen from the qualified candidates"), the original sample space of all possible outcomes is effectively reduced to only those outcomes that satisfy the given condition. The probability is then calculated within this new, smaller sample space.
• Working with Percentages: When a problem involves multiple percentages, assuming a convenient total number (e.g., 100 or 1000) for the initial population often simplifies calculations by converting percentages into whole numbers, making it easier to track subsets.

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Step-by-Step Solution

To simplify calculations involving percentages, let's assume a total of 100 candidates appeared for the exam. This allows us to convert percentages directly into concrete counts without dealing with decimals until the final probability calculation.

Step 1: Determine the Initial Number of Male and Female Candidates

• Why this step? We first need to establish the initial distribution of candidates by gender. This will be the basis for calculating subsets of candidates later.
• Let the total number of candidates be $100$.
• Number of female candidates: $60\%$ of $100 = 0.60 \times 100 = 60$.
• Number of male candidates: $40\%$ of $100 = 0.40 \times 100 = 40$.

Step 2: Calculate the Total Number of Qualified Candidates

• Why this step? The problem states that $60\%$ of all candidates qualify. This calculation gives us the size of our new, reduced sample space – the total group from which we will eventually pick a candidate.
• Total candidates qualifying the exam: $60\%$ of the total candidates.
• Number of qualified candidates = $60\%$ of $100 = 0.60 \times 100 = 60$.
  • So, out of the initial 100 candidates, 60 successfully qualified. This group of 60 is our new, effective sample space for the final probability calculation.

Step 3: Determine the Number of Qualified Females and Qualified Males

• Why this step? The problem provides a critical relationship about the composition of the qualified candidates: "The number of females qualifying the exam is twice the number of males qualifying it." We need to use this relationship, combined with the total number of qualified candidates found in Step 2, to determine the exact count of qualified females and males.
• Let $N_{QF}$ be the number of qualified female candidates.
• Let $N_{QM}$ be the number of qualified male candidates.
• From Step 2, we know that the total number of qualified candidates is $60$. This gives us our first equation: $N_{QF} + N_{QM} = 60 \quad \text{(Equation 1)}$
• The problem statement gives us the second relationship: "The number of females qualifying the exam is twice the number of males qualifying it." $N_{QF} = 2 \times N_{QM} \quad \text{(Equation 2)}$
• Now, we can solve this system of two linear equations. Substitute Equation 2 into Equation 1: $(2 \times N_{QM}) + N_{QM} = 60$ $3 \times N_{QM} = 60$
• Solve for $N_{QM}$: $N_{QM} = \frac{60}{3} = 20$
• Now, substitute the value of $N_{QM}$ back into Equation 2 to find $N_{QF}$: $N_{QF} = 2 \times 20 = 40$
• Thus, out of the $60$ qualified candidates, $40$ are females and $20$ are males.

Step 4: Calculate the Required Probability

• Why this step? We have now successfully identified our reduced sample space (the $60$ qualified candidates) and the number of favorable outcomes within that space (the $40$ qualified females). We can directly apply the definition of conditional probability from our key concepts.
• We need to find the probability that a randomly chosen candidate from the qualified candidates is a female. This is $P(\text{Female} | \text{Qualified})$.
• Using the conditional probability formula based on counts: $P(\text{Female} | \text{Qualified}) = \frac{\text{Number of Qualified Females}}{\text{Total Number of Qualified Candidates}}$ $P(\text{Female} | \text{Qualified}) = \frac{N_{QF}}{N_{QF} + N_{QM}}$ $P(\text{Female} | \text{Qualified}) = \frac{40}{60}$
• Simplifying the fraction by dividing both numerator and denominator by their greatest common divisor (20): $P(\text{Female} | \text{Qualified}) = \frac{2}{3}$

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Common Mistakes & Tips

• Confusing Joint and Conditional Probability: A frequent error is to calculate $P(\text{Female AND Qualified})$ (which would be $40/100 = 2/5$ from the initial pool) instead of $P(\text{Female | Qualified})$. The phrase "chosen from the qualified candidates" explicitly indicates conditional probability and a reduced sample space.
• Incorrectly Defining the Sample Space: Always identify the "given" condition to correctly determine the denominator for your probability calculation. In this case, the sample space is only the qualified candidates.
• Simplifying Calculations with a Base Value: When percentages are involved, assuming a convenient total number (like 100 or 1000) for the initial population can make the intermediate calculations with whole numbers much easier and less prone to decimal errors.

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Summary

This problem is a straightforward application of conditional probability. We started by assuming a total number of candidates (100) to convert percentages into concrete counts. Next, we determined the total number of qualified candidates (60), which became our reduced sample space. Using the given ratio that the number of qualified females is twice the number of qualified males, we calculated that there are 40 qualified females and 20 qualified males. Finally, the probability of choosing a female from this reduced sample space of qualified candidates was found by dividing the number of qualified females by the total number of qualified candidates.

The final answer is $\boxed{\frac{2}{3}}$, which corresponds to option (A).