Mutually exclusive vs independentपरस्पर अपवर्जी बनाम स्वतंत्रMutually exclusive vs independent

NCERT Remark (ii), Example 10 · Class 12 Maths · Chapter 13 · interactive diagram

Parts and what they mean

• Sample space of one die (एक पासे की प्रतिदर्श समष्टि) — S = \{1, 2, 3, 4, 5, 6\}, each outcome with probability \tfrac{1}{6}.
• E = number less than 3 (E = 3 से छोटी संख्या) — E = \{1, 2\}, P(E) = \tfrac{1}{3}.
• F = number more than 4 (F = 4 से बड़ी संख्या) — F = \{5, 6\}, P(F) = \tfrac{1}{3}. E and F have no common outcome.
• Mutually exclusive ⇒ dependent here (परस्पर अपवर्जी ⇒ यहाँ पराश्रित) — If E happens, F surely cannot: P(F|E) = 0 \ne P(F). Mutually exclusive events with nonzero probabilities are never independent.
• Sample space of one die (एक पासे की प्रतिदर्श समष्टि) — The same die, used for Example 10.
• E = multiple of 3 (E = 3 का गुणज) — E = \{3, 6\}, P(E) = \tfrac{1}{3}.
• F = even number (F = सम संख्या) — F = \{2, 4, 6\}, P(F) = \tfrac{1}{2}.
• E ∩ F = {6} (E ∩ F = {6}) — They share the outcome 6 — independent events CAN have common outcomes.
• Independent: product test holds (स्वतंत्र: गुणनफल परीक्षण सत्य) — P(E \cap F) = \tfrac{1}{6} = \tfrac{1}{3} \times \tfrac{1}{2} = P(E)\,P(F). Also P(E|F) = \tfrac{1}{3} = P(E).

Original diagram drawn by Simple Learning Hub, based on NCERT Remark (ii), Example 10. Report a mistake