Advanced Probability Analysis

🎲 Advanced Probability Analysis 16-in-1

📊 distributions · combinatorics · probability rules · Bayes · hypothesis tests · more

🎲 Binomial Distribution

n (trials) p (probability) k (successes)

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PMF, CDF, mean, variance

📊 Poisson Distribution

λ (mean) k (events)

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PMF, CDF, mean, variance

🔔 Normal Distribution

Mean (μ) Std Dev (σ) X value

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PDF, CDF, Z-score, percentile

📏 Z-Score & Percentile

Raw Score (x) Mean (μ) Std Dev (σ)

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z-score, percentile, interpretation

🔢 Combinations (nCr)

n (total items) r (choose)

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Number of ways to choose r from n

🔀 Permutations (nPr)

n (total items) r (arrange)

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Number of ways to arrange r from n

📊 Probability of A ∪ B

P(A) P(B) P(A ∩ B)

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P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

📊 Conditional Probability

P(A) P(B|A)

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P(A ∩ B) = P(A) × P(B|A)

🧠 Bayes' Theorem

P(A) P(B|A) P(B|¬A)

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P(A|B) = P(A) × P(B|A) / [P(A)P(B|A) + P(¬A)P(B|¬A)]

📊 Expected Value

Outcomes (comma separated) Probabilities (comma separated)

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E(X) = Σ x·P(x), variance, std dev

📊 Law of Total Probability

P(A|B₁) P(B₁) P(A|B₂) P(B₂)

—

P(A) = Σ P(A|Bᵢ)·P(Bᵢ)

📊 Independent Events

P(A) P(B)

—

P(A ∩ B), P(A ∪ B), P(A|B), P(B|A)

🎯 Geometric Distribution

p (success probability) k (first success on trial)

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P(X=k) = (1-p)^(k-1)·p, CDF, mean, variance

🎯 Hypergeometric Distribution

N (population size) K (successes in population) n (sample size) k (successes in sample)

—

PMF, CDF, mean, variance

⏱️ Exponential Distribution

λ (rate parameter) x (time)

—

PDF = λe^(-λx), CDF = 1-e^(-λx), mean, variance

📊 Chebyshev's Inequality

Mean (μ) Std Dev (σ) k (number of std deviations)

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P(|X-μ| ≥ kσ) ≤ 1/k² · bounds

🎲 Advanced Probability Analysis – 16-in-1
This comprehensive tool provides essential probability functions:

1. Binomial Distribution – PMF, CDF, mean, variance.
2. Poisson Distribution – PMF, CDF, mean, variance.
3. Normal Distribution – PDF, CDF, Z-score, percentile.
4. Z-Score & Percentile – convert raw scores to z-scores.
5. Combinations (nCr) – ways to choose r from n.
6. Permutations (nPr) – ways to arrange r from n.
7. Probability of A ∪ B – union with intersection.
8. Conditional Probability – P(A ∩ B) = P(A) × P(B|A).
9. Bayes' Theorem – update probabilities with evidence.
10. Expected Value – E(X), variance, std dev.
11. Law of Total Probability – P(A) = Σ P(A|Bᵢ)·P(Bᵢ).
12. Independent Events – intersection, union, conditional.
13. Geometric Distribution – first success probability.
14. Hypergeometric Distribution – sampling without replacement.
15. Exponential Distribution – waiting time distribution.
16. Chebyshev's Inequality – probability bounds.

💡 Tips:
• All probabilities must be between 0 and 1.
• All calculations are performed client-side – no data is sent anywhere.
• Perfect for students, researchers, and data analysts.

📖 Advanced Probability Analysis – detailed guide

🎲 Binomial Distribution
Models number of successes in n independent trials. PMF: P(X=k) = C(n,k)·pᵏ·(1-p)ⁿ⁻ᵏ. Mean = np, variance = np(1-p).

📊 Poisson Distribution
Models rare events. PMF: P(X=k) = e⁻λ·λᵏ/k!. Mean = λ, variance = λ.

🔔 Normal Distribution
The bell curve. PDF: f(x) = 1/(σ√2π)·e^(-(x-μ)²/(2σ²)). CDF gives probabilities.

📏 Z-Score & Percentile
z = (x - μ) / σ. The percentile is the probability of being less than or equal to z.

🔢 Combinations (nCr)
Number of ways to choose r items from n without order: n!/(r!(n-r)!).

🔀 Permutations (nPr)
Number of ways to arrange r items from n with order: n!/(n-r)!.

📊 Probability Rules
• Union: P(A∪B) = P(A) + P(B) - P(A∩B)
• Conditional: P(A∩B) = P(A)·P(B|A)
• Bayes: P(A|B) = P(A)P(B|A) / [P(A)P(B|A) + P(¬A)P(B|¬A)]
• Total Probability: P(A) = Σ P(A|Bᵢ)P(Bᵢ)
• Independence: P(A∩B) = P(A)P(B)

🎯 Geometric Distribution
Models number of trials until first success. PMF: P(X=k) = (1-p)ᵏ⁻¹p. Mean = 1/p, variance = (1-p)/p².

🎯 Hypergeometric Distribution
Models sampling without replacement. PMF: P(X=k) = C(K,k)·C(N-K,n-k)/C(N,n).

⏱️ Exponential Distribution
Models waiting time. PDF: f(x) = λe⁻λx, CDF: F(x) = 1-e⁻λx. Mean = 1/λ, variance = 1/λ².

📊 Chebyshev's Inequality
For any distribution: P(|X-μ| ≥ kσ) ≤ 1/k². Provides a lower bound on probability within k standard deviations.

⚡ All tools work offline · perfect for students and researchers

✅ advanced probability analysis · 16-in-1 · dynamic · no server