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Example 36 - A and B throw a die alternatively till one of them - Examples

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  1. Chapter 13 Class 12 Probability
  2. Serial order wise
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` Example 36 A and B throw a die alternatively till one of them gets a ‘6’ and wins the game. Find their respective probabilities of winning, if A starts first. Winning the game is getting a 6 on the die P(getting 6) = 1﷮6﷯ P(not getting six) = 1 – P(getting six) = 1 – 1﷮6﷯ = 5﷮6﷯ 1st throw by A: A gets a six P(A wins) = 1﷮6﷯ 2nd throw by B: A does not get 6, B gets six So, P(B wins) = 5﷮6﷯ × 1﷮6﷯ 3rd throw by A: A does not get 6, B does not get 6, A gets six P(A wins) = 5﷮6﷯ × 5﷮6﷯ × 1﷮6﷯ 4th throw by B: A does not get 6, B does not get 6, A does not get 6, B gets six P(B wins) = 5﷮6﷯ × 5﷮6﷯ × 5﷮6﷯ × 1﷮6﷯ 5th throw by A: A does not get 6, B does not get 6, A does not get 6, B does not get 6, A gets six P(A wins) = 5﷮6﷯ × 5﷮6﷯ × 5﷮6﷯ × 5﷮6﷯ × 1﷮6﷯ and so on So, probability that A wins is P(A wins) = 1﷮6﷯ + 5﷮6﷯ × 5﷮6﷯ × 1﷮6﷯ + 5﷮6﷯ × 5﷮6﷯ × 5﷮6﷯ × 5﷮6﷯ × 1﷮6﷯ + ……. = 1﷮6﷯﷯ + 5﷮6﷯﷯﷮2﷯ 1﷮6﷯﷯ + 5﷮6﷯﷯﷮4﷯ 1﷮6﷯﷯ + ............ = 1﷮6﷯﷮1 − 5﷮6﷯﷯﷮2﷯﷯ = 1﷮6﷯﷮1 − 25﷮36﷯﷯ = 1﷮6﷯﷮ 36 − 25﷮36﷯﷯ = 1﷮6﷯﷮ 11﷮36﷯﷯ = 6﷮11﷯ ∴ P(A wins) = 𝟔﷮𝟏𝟏﷯ and P(B wins) = 1 – P(A wins) = 1 – 6﷮11﷯ = 𝟓﷮𝟏𝟏﷯

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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