Partition of S and the theorem of total probabilityS का विभाजन और संपूर्ण प्रायिकता की प्रमेयS ka partition aur total probability ka theorem
NCERT Fig 13.4 · Class 12 Maths · Chapter 13 · interactive diagram
Parts and what they mean
- E₁ — one piece of the partition (E₁ — विभाजन का एक टुकड़ा) — The pieces E_1, \ldots, E_4 do not overlap, together they fill S, and each has P(E_1) > 0.
- A ∩ E₁ (A ∩ E₁) — The part of A inside E_1. By the multiplication rule P(A \cap E_1) = P(E_1)\,P(A|E_1).
- E₂ — one piece of the partition (E₂ — विभाजन का एक टुकड़ा) — The pieces E_1, \ldots, E_4 do not overlap, together they fill S, and each has P(E_2) > 0.
- A ∩ E₂ (A ∩ E₂) — The part of A inside E_2. By the multiplication rule P(A \cap E_2) = P(E_2)\,P(A|E_2).
- E₃ — one piece of the partition (E₃ — विभाजन का एक टुकड़ा) — The pieces E_1, \ldots, E_4 do not overlap, together they fill S, and each has P(E_3) > 0.
- A ∩ E₃ (A ∩ E₃) — The part of A inside E_3. By the multiplication rule P(A \cap E_3) = P(E_3)\,P(A|E_3).
- E₄ — one piece of the partition (E₄ — विभाजन का एक टुकड़ा) — The pieces E_1, \ldots, E_4 do not overlap, together they fill S, and each has P(E_4) > 0.
- A ∩ E₄ (A ∩ E₄) — The part of A inside E_4. By the multiplication rule P(A \cap E_4) = P(E_4)\,P(A|E_4).
- Event A (घटना A) — A is cut by the partition into disjoint pieces A \cap E_1, \ldots, A \cap E_4.
- Sample space S (प्रतिदर्श समष्टि S) — S = E_1 \cup E_2 \cup E_3 \cup E_4.
- Theorem of total probability (संपूर्ण प्रायिकता की प्रमेय) — Add the pieces of A: P(A) = \sum_{j} P(E_j)\,P(A|E_j). Bayes then takes one piece over this total.
Original diagram drawn by Simple Learning Hub, based on NCERT Fig 13.4. Report a mistake