Class CBSE Class 12 Mathematics Probability Q #1303
COMPETENCY BASED
UNDERSTAND
3 Marks 2024 AISSCE(Board Exam) SA
E and F are two independent events such that $P(\overline{E})=0\cdot6$ and $P(E\cup F)=0\cdot6$ Find $P(F)$ and $P(\overline{E}\cup\overline{F})$
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Detailed Solution

Step 1: Find $P(E)$

Given $P(\overline{E}) = 0.6$, we can find $P(E)$ using the complement rule: $P(E) = 1 - P(\overline{E}) = 1 - 0.6 = 0.4$

Step 2: Use the formula for $P(E \cup F)$

We are given $P(E \cup F) = 0.8$. We also know that for any two events $E$ and $F$: $P(E \cup F) = P(E) + P(F) - P(E \cap F)$

Step 3: Use independence to find $P(E \cap F)$

Since $E$ and $F$ are independent events, $P(E \cap F) = P(E)P(F)$. Substituting this into the previous equation: $P(E \cup F) = P(E) + P(F) - P(E)P(F)$

Step 4: Solve for $P(F)$

Plugging in the known values: $0.8 = 0.4 + P(F) - 0.4P(F)$ $0.8 - 0.4 = P(F)(1 - 0.4)$ $0.4 = 0.6P(F)$ $P(F) = \frac{0.4}{0.6} = \frac{2}{3}$

Step 5: Find $P(\overline{E} \cup \overline{F})$

Using De Morgan's Law: $P(\overline{E} \cup \overline{F}) = P(\overline{E \cap F}) = 1 - P(E \cap F)$ Since $E$ and $F$ are independent, $P(E \cap F) = P(E)P(F) = 0.4 \times \frac{2}{3} = \frac{0.8}{3} = \frac{4}{15}$ Therefore, $P(\overline{E} \cup \overline{F}) = 1 - \frac{4}{15} = \frac{15 - 4}{15} = \frac{11}{15}$

Final Answer: $P(F) = \frac{2}{3}$ and $P(\overline{E}\cup\overline{F}) = \frac{11}{15}$

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Pedagogical Audit
Bloom's Analysis: This is an UNDERSTAND question because the student needs to understand the concepts of independent events, complement rule, and De Morgan's law to solve the problem. They must also understand how to apply the formula for the probability of the union of two events.
Knowledge Dimension: CONCEPTUAL
Justification: The question requires understanding of concepts like independent events, probability of union of events, and De Morgan's law. It's not just about recalling facts or applying a specific algorithm.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. The question requires the student to apply their knowledge of probability to solve a problem involving independent events and De Morgan's law.

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