Probability Case Study Class 12 PDF
Probability case study class 12 pdf helps students understand how probability concepts apply to real-life situations. Moreover, these questions follow the latest CBSE competency-based exam pattern. Students develop analytical thinking through structured problems. Therefore, regular practice improves confidence. Short calculations save time.
Topics Covered in Probability Case Studies
Case study questions include events, outcomes, and conditional probability. For example, students analyze cards, dice, or data-based situations. As a result, they apply formulas logically. In addition, these problems strengthen interpretation skills. Thus, conceptual clarity improves steadily.
How to Use PDF Study Material Effectively
Firstly, revise probability formulas thoroughly. Then, practice solved case study questions step by step. Meanwhile, focus on understanding the given data. Additionally, reviewing similar questions boosts accuracy. Consequently, students perform better in Class 12 board exams.
Case Study 4: Probability case study class 12 pdf
Case Study Description: A toy factory manufactures a specific small electronic component in batches. Historically, the factory has an **average defect rate of $10\%$** for this component. The quality control manager decides to implement a detailed inspection process. During this process, inspectors are instructed to randomly select 4 components from a batch and test them for defects. The number of defective components found in the sample of 4 is the **Random Variable** $X$.
The factory manager is interested in analyzing the distribution of $X$, specifically the probability of finding 0, 1, 2, 3, or 4 defective components. This scenario represents a classic application of the **Binomial Distribution** since the trials (testing each component) are independent, there are a fixed number of trials ($n=4$), and the probability of a success (finding a defective component, $p=0.10$) remains constant. Understanding the **probability distribution** is vital for setting acceptable quality benchmarks. The manager also needs to calculate the **mean** (expected number of defects) and the **variance** of this random variable $X$ to predict long-term quality trends and assess risk. For this analysis, we assume the batch size is very large, making the selection of components approximately independent, which satisfies the conditions for a Binomial distribution.
Let $X \sim B(n, p)$ where $n=4$ and $p=0.10$. Probability of non-defective is $q=0.90$.
Theory and Formulae Related to Binomial Distribution:
- **Probability Mass Function**: $P(X=k) = \binom{n}{k} p^k q^{n-k}$
- **Mean (Expected Value)**: $E(X) = n \cdot p$
- **Variance**: $\text{Var}(X) = n \cdot p \cdot q$
- **Complementary Event**: $P(X \ge 1) = 1 – P(X=0)$
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