The problem states that two players, A and B, are playing a game where they alternate rolling a fair dice. The objective of the game is to be the first player to roll a six. Player A starts the game.


To find the probability that player A wins the game, we can consider the different possible outcomes of the game and calculate the probability of each outcome.


There are several possible outcomes for the game:
1. Player A rolls a six on the first roll and wins the game.
2. Player A rolls a number other than six, then player B rolls a number other than six, and so on, until player A eventually rolls a six and wins the game.
3. Player A rolls a number other than six, then player B rolls a six and wins the game.


Let's calculate the probability of each outcome:

1. Player A rolls a six on the first roll and wins the game:
The probability of this outcome is simply 1/6, as there is only one way for player A to roll a six on the first roll.

2. Player A rolls a number other than six, then player B rolls a number other than six, and so on, until player A eventually rolls a six and wins the game:
This outcome can be represented as an infinite geometric series. The probability of player A rolling a number other than six on each roll is 5/6, and the probability of player B rolling a number other than six on each roll is also 5/6. Therefore, the probability of this outcome can be calculated using the formula for the sum of an infinite geometric series:

P = (5/6)^2 + (5/6)^4 + (5/6)^6 + ...

Using the formula for the sum of an infinite geometric series, we can simplify this expression:

P = (5/6)^2 * (1 + (5/6)^2 + (5/6)^4 + ...)
P = (5/6)^2 * (1/(1 - (5/6)^2))
P = (5/6)^2 * (1/(1 - 25/36))
P = (5/6)^2 * (1/(11/36))
P = (5/6)^2 * (36/11)
P = 25/66

3. Player A rolls a number other than six, then player B rolls a six and wins the game:
The probability of player A rolling a number other than six on each roll is 5/6, and the probability of player B rolling a six on each roll is 1/6. Therefore, the probability of this outcome is (5/6) * (1/6) = 5/36.


To calculate the probability of player A winning, we need to sum up the probabilities of the first and second outcomes, as the third outcome represents a win for player B.

P(A wins) = P(outcome 1) + P(outcome 2)
P(A wins) = 1/6 + 25/66
P(A wins) = 11/66 + 25/66
P(A wins) =