Probability
Class 11 me basic probability dekhi. Ab usse aage badhte hain — conditional probability, independence, Bayes' theorem, aur random variables. Ye topics statistics aur machine learning tak ki neev hain.
1. Conditional Probability
Event \(A\) ki probability, jab pata ho ki \(B\) ho chuka hai: \[ P(A \mid B) = \frac{P(A \cap B)}{P(B)}, \qquad P(B) \neq 0. \]
Ek dice phenka. Diya gaya hai ki even aaya (\(B = \{2,4,6\}\)). Number \(4\) hone ki probability? \[ P(A \mid B) = \frac{P(\{4\})}{P(\{2,4,6\})} = \frac{1/6}{3/6} = \frac{1}{3}. \]
2. Multiplication Rule
\[ P(A \cap B) = P(A)\,P(B \mid A) = P(B)\,P(A \mid B). \]
3. Independent Events
Do events independent hain agar ek ke hone se doosre ki probability par koi farak na pade: \[ P(A \cap B) = P(A)\,P(B). \]
(Iske liye \(P(A \mid B) = P(A)\) bhi hota hai.) Dhyan: independent aur mutually exclusive alag cheezein hain.
4. Theorem of Total Probability
Agar \(E_1, E_2, \dots, E_n\) mutually exclusive aur exhaustive events ho, to kisi event \(A\) ke liye:
\[ P(A) = \sum_{i=1}^{n} P(E_i)\,P(A \mid E_i). \]
5. Bayes' Theorem
Agar \(A\) ho gaya hai, to wo kisi particular cause \(E_i\) se hone ki probability: \[ P(E_i \mid A) = \frac{P(E_i)\,P(A \mid E_i)}{\displaystyle\sum_{j} P(E_j)\,P(A \mid E_j)}. \]
Ye "ulti" probability deta hai — result se cause ki taraf. Diagnostic tests, spam filters sab isi par chalte hain.
Do bags: Bag 1 me 3 red, 2 blue; Bag 2 me 1 red, 4 blue. Ek bag randomly chuna (\(P = \frac12\) each), usse ek red ball nikla. Wo Bag 1 se hone ki probability? \[ P(B_1 \mid R) = \frac{\frac12\cdot\frac35}{\frac12\cdot\frac35 + \frac12\cdot\frac15} = \frac{3/10}{4/10} = \frac{3}{4}. \]
6. Random Variable aur Expectation
- Random variable \(X\): har outcome ko ek number assign karta hai.
- Probability distribution: har value \(x_i\) ke saath uski probability \(p_i\); aur \(\sum p_i = 1\).
- Mean (Expectation): \(E(X) = \sum x_i\,p_i\).
- Variance: \(\operatorname{Var}(X) = \sum x_i^{2}\,p_i - [E(X)]^{2}\).
Key Takeaways
- \(P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}\).
- Independent: \(P(A \cap B) = P(A)P(B)\) (mutually exclusive se alag).
- Total probability se overall \(P(A)\); Bayes' theorem se cause ki probability.
- Random variable: \(E(X) = \sum x_i p_i\), \(\operatorname{Var}(X) = \sum x_i^{2}p_i - [E(X)]^{2}\).