Class CBSE Class 12 Mathematics Probability Q #1369
COMPETENCY BASED
UNDERSTAND
3 Marks 2025 AISSCE(Board Exam) SA
For the vacancy advertised in the newspaper, 3000 candidates submitted their applications. From the data it was revealed that two third of the total applicants were females and other were males. The selection for the job was done through a written test. The performance of the applicants indicates that the probability of a male getting a distinction in written test is 0.4 and that a female getting a distinction is 0.35. Find the probability that the candidate chosen at random will have a distinction in the written test.
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Detailed Solution

Step 1: Define Events

Let $E_1$ be the event that the candidate is female, and $E_2$ be the event that the candidate is male. Let $A$ be the event that the candidate gets a distinction in the written test.

Step 2: Calculate Probabilities of $E_1$ and $E_2$

Given that two-thirds of the applicants are females, $P(E_1) = \frac{2}{3}$. Since the rest are males, $P(E_2) = 1 - P(E_1) = 1 - \frac{2}{3} = \frac{1}{3}$.

Step 3: Define Conditional Probabilities

The probability of a male getting a distinction is $P(A|E_2) = 0.4$, and the probability of a female getting a distinction is $P(A|E_1) = 0.35$.

Step 4: Apply the Law of Total Probability

We want to find the probability that a candidate chosen at random will have a distinction in the written test, which is $P(A)$. Using the law of total probability, we have: $$P(A) = P(A|E_1)P(E_1) + P(A|E_2)P(E_2)$$

Step 5: Calculate $P(A)$

Substitute the values we have: $$P(A) = (0.35)\left(\frac{2}{3}\right) + (0.4)\left(\frac{1}{3}\right) = \frac{0.7}{3} + \frac{0.4}{3} = \frac{1.1}{3} = \frac{11}{30}$$

Step 6: Final Answer

Therefore, the probability that the candidate chosen at random will have a distinction in the written test is $\frac{11}{30}$.

Final Answer: 11/30

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Pedagogical Audit
Bloom's Analysis: This is an UNDERSTAND question because the student needs to understand the problem, identify the relevant probabilities, and apply the law of total probability to find the solution.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to apply a specific procedure (law of total probability) to solve the problem. The student needs to know how to use the formula and apply it to the given data.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. The question is designed to test the student's ability to apply the concepts of probability to a real-world scenario, rather than just recalling definitions or formulas.

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