Answer: To find the probability that Alice will win the game, we can use a probability tree diagram or a simpler approach by considering the possible outcomes.
There are two ways Alice can win the game:
Alice rolls a six on her first turn.
Alice rolls something other than a six on her first turn, and then Ben also rolls something other than a six on his first turn, and so on until Alice eventually rolls a six.
Let's calculate the probability of each scenario:
The probability that Alice rolls a six on her first turn is 1/6, as there is only one favorable outcome (rolling a six) out of six possible outcomes (rolling a die).
The probability that Alice and Ben both roll something other than a six on their first turns is (5/6) * (5/6) = 25/36. This is because there are 5 possible outcomes (rolling a number other than six) out of 6 possible outcomes for each roll.
If Alice and Ben both roll something other than a six on their first turns, the game resets and the probability of Alice eventually rolling a six remains the same. The probability of this happening is the same as the probability of Alice winning the entire game from the beginning.
So, the overall probability that Alice will win the game can be calculated as follows:
Probability(Alice wins) = Probability(Alice rolls a six on her first turn) + Probability(Alice and Ben both roll something other than a six) * Probability(Alice wins from the beginning)
Probability(Alice wins) = (1/6) + (25/36) * Probability(Alice wins)
Now, let's solve for Probability(Alice wins):
Probability(Alice wins) - (25/36) * Probability(Alice wins) = 1/6
(1 - 25/36) * Probability(Alice wins) = 1/6
(11/36) * Probability(Alice wins) = 1/6
Probability(Alice wins) = (1/6) / (11/36)
To divide by a fraction, we multiply by its reciprocal:
Probability(Alice wins) = (1/6) * (36/11)
Multiply the numerators and denominators:
Probability(Alice wins) = 36/66
Now, simplify the fraction by finding the greatest common divisor (GCD) of 36 and 66, which is 6:
Probability(Alice wins) = (36/6) / (66/6) = 6/11
So, the probability that Alice will win the game is 6/11.