Probability Advance
Conditional Probability, Law of Total Probability
Section titled “Conditional Probability, Law of Total Probability”Probability of event A given that B has occurred.
Meaning (Intuition)
- Sample space shrinks to B
- We ask: within B, how often does A occur?
Rearranged Forms (Very Useful ⭐)
If A and B are independent:
Law of Total Probability ⭐⭐
🔹 Statement
Section titled “🔹 Statement”If are mutually exclusive and exhaustive events, then for any event B:
P(B) = \sum_{i=1}^{n} P(B \mid A_i)P(A_i) }$$ **Why this works (Reason)** - Event B can occur **only through one of the ($A_i$)** - Break B into disjoint parts: $B = (B \cap A_1) \cup (B \cap A_2) \cup \dots$ **Diagram View (Mental)** ``` A1 ─┐ A2 ─┼──► B A3 ─┘ ``` ##### **Quick Comparison Table** | Concept | Formula | Purpose | | ----------------------- | ---------------------------------------------------- | ------------------- | | Conditional Probability | $P(A\mid B)$ | Probability after B | | Joint Probability | $P(A\cap B)$) | Both occur | | Total Probability | $\sum P(B\mid A_i)P(A_i)$ | Find $P(B)$ | | Bayes’ Theorem | $\frac{P(B\mid A_i)P(A_i)}{\sum P(B\mid A_k)P(A_k)}$ | Reverse condition | **Exam Memory Tricks 🧠** - **Conditional** → _Restrict sample space_ - **Total probability** → _Weighted sum_ - **Bayes** → _Reverse the condition_ - **Independent?** → conditioning doesn’t change probability **When to Use What?** - Given **“given that”** → Conditional probability - Given **multiple sources/causes** → Total probability - Asked **cause after effect** → Bayes’ theorem ## **Bayes’ Theorem** Bayes’ theorem gives the **probability of a cause given an observed event**. ##### **General Form (Multiple Causes)** ⭐ If $A_1, A_2, \dots, A_n$ are **mutually exclusive and exhaustive events**, then: $$\boxed{ P(A_i \mid B) = \frac{P(B \mid A_i)P(A_i)} {\sum_{k=1}^{n} P(B \mid A_k)P(A_k)} } $$ ##### **Standard Two-Event Case** For events AAA and its complement $A'$: $${ P(B) = P(B \mid A)P(A) + P(B \mid A')P(A') }$$ $$\boxed{ P(A \mid B) = \frac{P(B \mid A)P(A)} {P(B \mid A)P(A) + P(B \mid A')P(A')} } $$ ##### **Why Bayes’ Theorem Works (Intuition)** Using **Conditional Probability** $$P(A \mid B) = \frac{P(A \cap B)}{P(B)}, \quad (P(B) > 0)$$ - But by definition of conditional probability, $$P(A \cap B) = P(B \mid A)P(A)$$ - Substitute into the first equation: $$\boxed{P(A_i \mid B) = \frac{P(B \mid A_i)P(A_i)}{P(B)}}$$ **Using Law of Total Probability**P(B) = \sum_{i=1}^{n} P(B \mid A_i)P(A_i)
- Substitute this value of $P(B)$ into Bayes’ formula: $$\boxed{ P(A_i \mid B) = \frac{P(B \mid A_i)P(A_i)} {\sum_{k=1}^{n} P(B \mid A_k)P(A_k)} } $$ **Meaning of Each Term** | Term | Meaning | | --------------- | ----------------------------------------- | | $P(A_i)$ | Prior probability (before seeing $B$) | | $P(B \mid A_i)$ | Likelihood | | $P(B)$ | Total probability of (B) | | $P(A_i \mid B)$ | Posterior probability | **Posterior ∝ Likelihood × Prior** **Common GATE Traps** 1. Confusing $(P(A \mid B))$ with $(P(B \mid A))$ 2. Forgetting denominator (total probability) 3. Not checking **mutually exclusive & exhaustive** 4. Using Bayes when **simple conditional probability** is enough