Multiplication rule on a tree (Example 8)वृक्ष पर गुणन नियम (उदाहरण 8)Tree par multiplication rule (Example 8)

NCERT Example 8 · Class 12 Maths · Chapter 13 · interactive diagram

Parts and what they mean

• Urn: 10 black, 5 white (कलश: 10 काली, 5 सफ़ेद) — Two balls are drawn one after the other, without replacement.
• First draw (पहली निकासी) — Black with probability \tfrac{10}{15}, white with \tfrac{5}{15} (they add to 1).
• First ball black (पहली गेंद काली) — Now 14 balls are left: 9 black and 5 white.
• First ball white (पहली गेंद सफ़ेद) — Now 14 balls are left: 10 black and 4 white.
• Second draw — conditional (दूसरी निकासी — सप्रतिबंध) — Each second branch is a conditional probability like P(F|E): it depends on what the first draw took out.
• Path: black, then black (रास्ता: काली, फिर काली) — P(E \cap F) = P(E)\,P(F|E) = \tfrac{10}{15} \times \tfrac{9}{14} = \tfrac{3}{7} — multiply along the path.
• Both black (दोनों काली) — \tfrac{10}{15} \times \tfrac{9}{14} = \tfrac{90}{210} = \tfrac{3}{7} — the answer of Example 8.
• Black then white (काली फिर सफ़ेद) — \tfrac{10}{15} \times \tfrac{5}{14} = \tfrac{5}{21}
• White then black (सफ़ेद फिर काली) — \tfrac{5}{15} \times \tfrac{10}{14} = \tfrac{5}{21}
• Both white (दोनों सफ़ेद) — \tfrac{5}{15} \times \tfrac{4}{14} = \tfrac{2}{21}
• Tree check (वृक्ष-जाँच) — All paths together are certain: \tfrac{3}{7} + \tfrac{5}{21} + \tfrac{5}{21} + \tfrac{2}{21} = 1.

Original diagram drawn by Simple Learning Hub, based on NCERT Example 8. Report a mistake