Conditional.html

---
title: Conditional Probability
layout: default
permalink: /Conditional.html
---

<header>
  <span class="badge">Probability</span>
  <h1>Conditional Probability (조건부 확률)</h1>
</header>

<section class="card">
  <div class="kicker">정의</div>
  <p class="pill">$P(A \mid B) = \dfrac{P(A \cap B)}{P(B)} \quad (P(B)>0)$</p>
  <ul>
    <li>$P(A\cap B)$: 동시에 발생</li>
    <li>$P(B)$: 조건 사건</li>
  </ul>
</section>

<section class="grid">
  <div class="card">
    <h2>1) 곱셈정리</h2>
    <p class="pill">$P(A\cap B)=P(A\mid B)P(B)=P(B\mid A)P(A)$</p>
  </div>
  <div class="card">
    <h2>2) 전확률법칙</h2>
    <p class="pill">$P(A)=\sum_i P(A\mid B_i)P(B_i)$</p>
  </div>
</section>

<section class="card">
  <h2>3) 베이즈 정리</h2>
  <p class="pill">$P(A \mid B) = \dfrac{P(B \mid A)P(A)}{P(B)}$</p>
  <p>$P(B)=\sum_i P(B \mid A_i)P(A_i)$</p>
  <p><a href="/Conditional-solutions.html">연습문제 해설 보기 →</a></p>
</section>

<section class="grid">
  <div class="card">
    <h2>4) 독립/조건부 독립</h2>
    <ul>
      <li>독립: $P(A\cap B)=P(A)P(B)$ ⇔ $P(A|B)=P(A)$</li>
      <li>조건부 독립: $P(A\cap B|C)=P(A|C)P(B|C)$</li>
    </ul>
  </div>
  <div class="card">
    <h2>5) 연속형</h2>
    <p class="pill">$f_{X|Y}(x|y)=\dfrac{f_{X,Y}(x,y)}{f_Y(y)}$</p>
  </div>
</section>

<section class="card">
  <h2>6) 미니 연습문제</h2>
  <ol>
    <li>주사위: $A$=짝수, $B$=$\ge4$. $P(A|B)$?</li>
    <li>이진 분류: 정답(+) 40%, $P(+|\hat{+})=0.8$, $P(-|\hat{-})=0.9$. $P(\hat{+})$?</li>
    <li>상자 A/B 문제: $P(B|\text{파란})$?</li>
  </ol>
</section>