#P05 · Probability, Statistics & AI · Lv.2View in 3D Mind-Map

Binomial Distribution

Jacob Bernoulli · 1713 CE

P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad X \sim B(n, p)

E[X] = np, \quad \operatorname{Var}(X) = np(1-p)

Models the probability of getting exactly k successes in n independent Bernoulli trials with success probability p.

🧮 Live Formula Calculator & Plotter

Trials n

Successes k

Success Prob p

P(X=4) = \binom{12}{4}(0.3)^{4}(0.70)^{8} = 23.11\%

확률 P(X = 4)

23.11%

기댓값 E[X] = np

3.60

표준편차 σ = √npq

1.59

Prerequisite Path (8 STEP)

Full Path Analyzer

Click steps in order to follow the 3D neon path from basics to target.

STEP 1#A01Linear Equation & Balance Axiom

ax + b = 0 \implies x = -\frac{b}{a}

STEP 2#A02Polynomial Expansion & Factoring

(a+b)^2 = a^2 + 2ab + b^2

STEP 3#A03Laws of Exponents & Radicals

a^m \cdot a^n = a^{m+n}

STEP 4#P01Permutations & Combinations

{}_nP_r = \frac{n!}{(n-r)!}

STEP 5#P02Conditional Probability & Chain Rule

P(A \mid B) = \frac{P(A \cap B)}{P(B)}

STEP 6#A08Binomial Theorem & Pascal's Triangle

(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

STEP 7#P04Expected Value & Variance

E[X] = \sum_i x_i p_i

STEP 8#P05Binomial Distribution

P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}

🚀 Next Unlocks (Derived Formulas) (1)

#P06Normal (Gaussian) Distribution & Z-Score

Geometric & Intuitive Insight

(p + q)^n을 이항정리로 전개했을 때 k번째 항이 곧 n번 중 k번 성공할 확률과 정확히 일치합니다.

Step-by-Step Derivation

Step 1.특정 순서로 k번 성공하고 (n-k)번 실패할 확률은 독립성에 의해 p^k (1-p)^(n-k)입니다.
Step 2.n번의 시행 중 성공이 일어날 k개의 위치를 고르는 방법의 수는 조합 C(n, k)가지입니다.
Step 3.따라서 P(X=k) = C(n, k) p^k (1-p)^(n-k)

AI & Coding Real-World Use

Used in ad CTR A/B testing, neural network Dropout masking probability modeling, and game gacha drop-rate simulation.

#A/B Testing#Dropout Regularization#CTR Modeling#Game Gacha Math

const binomPMF = (n, k, p) => nCr(n, k) * Math.pow(p, k) * Math.pow(1 - p, n - k);

Probability, Statistics & AI — Related Formulas in This Cluster

Permutations & Combinations

Lv.1 · Combinatorics

#P01

Conditional Probability & Chain Rule

Lv.2 · LLM Next-Token

Expected Value & Variance

Lv.2 · Reinforcement Learning

#P04

Normal (Gaussian) Distribution & Z-Score

Lv.3 · Diffusion Models

#P06

Central Limit Theorem & Confidence Intervals

Lv.4 · Mini-Batch SGD

#P07