Final answer:
A starts the game and has a probability of 1 of winning, while B has a probability of 0.
Step-by-step explanation:
To find the probabilities of A and B winning the game, we can consider the possible scenarios. Let's assume A starts the game.
Scenario 1: A wins on the first roll by getting a '6'.
Since the die has 6 sides, the probability of A winning on the first roll is 1/6.
Scenario 2: A doesn't win on the first roll, but B wins on the second roll by getting a '6'.
The probability of A not getting a '6' on the first roll is 5/6. And the probability of B getting a '6' on the second roll is 1/6. So, the probability of this scenario is (5/6) * (1/6).
We can continue this pattern for each subsequent roll until either A or B wins.
Therefore, the probability of A winning is the sum of the probabilities of all the scenarios where A wins, which can be written as:
P(A) = 1/6 + (5/6) * (1/6) + (5/6)^2 * (1/6) + ...
Simplifying the equation gives us:
P(A) = 1/6 * (1 + (5/6) + (5/6)^2 + ...)
This is an infinite geometric series where a = 1/6 and r = 5/6.
The formula for the sum of an infinite geometric series is:
Sum = a / (1 - r)
Substituting the values, we get:
P(A) = (1/6) / (1 - (5/6)) = 1/6 * 6/1 = 1
Therefore, the probability of A winning is 1.
Since this is a two-player game, the probability of B winning is 1 - P(A), which is 0.