Class CBSE Class 12 Mathematics Probability Q #1800
COMPETENCY BASED
APPLY
3 Marks 2026 AISSCE(Board Exam) SA
The probability of hitting the target by a trained sniper is three times the probability of not hitting the target on a stormy day due to high wind speed. The sniper fired two shots on the target on a stormy day when wind speed was very high. Find the probability that (i) target is hit (ii) atleast one shot misses the target.
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Detailed Solution

Step 1: Define Probabilities

Let $p$ be the probability of hitting the target and $q$ be the probability of missing the target. Given $p = 3q$. Since $p + q = 1$, we have $3q + q = 1$, which implies $4q = 1$. Thus, $q = 0.25$ and $p = 0.75$ (or $3/4$).

Step 2: Probability of hitting the target (at least once)

The sniper fires two shots. The probability of missing both shots is $q \times q = (1/4) \times (1/4) = 1/16$. The probability that the target is hit at least once is $1 - P(\text{missing both}) = 1 - 1/16 = 15/16$.

Step 3: Probability that at least one shot misses

The probability that at least one shot misses is the complement of the event that both shots hit the target. $P(\text{both hit}) = p \times p = (3/4) \times (3/4) = 9/16$. Therefore, $P(\text{at least one miss}) = 1 - 9/16 = 7/16$.

Final Answer: (i) 15/16, (ii) 7/16

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must translate a verbal condition into algebraic probability values and then apply the complement rule for independent events.
Knowledge Dimension: PROCEDURAL
Justification: The student follows a sequence of steps: solving for individual probabilities, identifying independent events, and applying the complement rule.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. This question tests the application of the Probability chapter (Bernoulli trials/Independent events) in a real-world scenario.

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