Probability(急)?

Question

題目: A bag contains 4 yellow balls, 2 green balls and 1 red ball. In a game, two players A and B take a ball alternately from the bag with replacement. The first player who takes the red ball wins the game. If player A takes a ball first, what is the probability that player A will win the game.

Answer 1

最愛解答

Question：

 A bag contains 4 yellow balls, 2 green balls and 1 red ball. In a game, two players A and B take a ball alternately from the bag with replacement. The first player who takes the red ball wins the game. If player A takes a ball first, what is the probability that player A will win the game.

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Solution：

The probability that the player takes a red ball in the first chance

= 1/(4 + 2 + 1)

= 1/7

The probability that the player takes a red ball in the second chance

= (1 - 1/7) (1 - 1/7) (1/7)

= (6/7)² (1/7)

The probability that the player takes a red ball in the third chance

= (1 - 1/7) (1 - 1/7) (1 - 1/7) (1 - 1/7) (1/7)

= (6/7)⁴ (1/7)

The probability that player A will win the game

= 1/7 + (6/7)² (1/7) + (6/7)⁴ (1/7) + (6/7)⁶ (1/7) + ...

= (1/7) / [1 - (6/7)²]

= 7/13

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Remark：

The sum to infinity of a geometric series = a/(1 - r) where

a is the first term,

r is the common ratio and -1 < r < 1.