Probability Distributions — Chapter Summary
Maharashtra Board · Class 12 · Mathematics & Statistics -Commerce
Summary of Probability Distributions for Maharashtra Board Class 12 Mathematics & Statistics -Commerce. The Binomial Distribution is one of the most.
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Overview
The Binomial Distribution is one of the most important discrete probability distributions in statistics. It deals with experiments that have exactly two outcomes (success or failure) and helps us calculate probabilities when we repeat such experiments multiple times. This distribution is widely used
Key Concepts
A Bernoulli trial is a single
A Bernoulli trial is a single experiment with exactly two outcomes: success (probability p) or failure (probability q = 1-p). Example: Tossing a coin
P(X = x) = ⁿCₓ pˣ
P(X = x) = ⁿCₓ pˣ qⁿ⁻ˣ where X is number of successes, n is total trials, p is probability of success, q = 1-p. Step-by-step application: (1) Identify
For binomial distribution X~B(n
For binomial distribution X~B(n,p): Mean μ = E(X) = np, Variance σ² = Var(X) = npq, Standard deviation σ = √(npq). These formulas provide quick ways t
P(X ≤ k) = sum
P(X ≤ k) = sum of P(X = 0) + P(X = 1) + ... + P(X = k). P(X ≥ k) = 1 - P(X ≤ k-1). P(X > k) = 1 - P(X ≤ k). Example: For P(X ≥ 6) in 10 coin tosses, c
Learning Objectives
- Understand and identify Bernoulli trials and their characteristics
- Derive and apply the binomial probability formula P(X = x) = ⁿCₓ pˣ qⁿ⁻ˣ
- Solve problems involving binomial distribution with step-by-step approach
- Calculate mean (μ = np) and variance (σ² = npq) of binomial distribution
- Apply binomial distribution to real-world problems like quality control, medical testing, and games of chance
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Content is aligned to the official syllabus. Refer to the board website for the latest curriculum.
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