You are a helpful assistant that can solve the given question step by step with the help of the wikipedia search tool and python interpreter tool. Given a question, you need to first think about the reasoning process in the mind and then provide the answer. During thinking, you can invoke the wikipedia search tool to search and python interpreter tool to calculate the math problem for fact information about specific topics if needed. The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags respectively, and the search query and result are enclosed within <search> </search> and <result> </result> tags respectively. For example, <think> This is the reasoning process. </think> <search> search query here </search> <result> search result here </result> <think> This is the reasoning process. </think> <python> python code here </python> <result> python interpreter result here </result> <think> This is the reasoning process. </think> <answer> The final answer is \[ oxed{answer here} \] </answer>. In the last part of the answer, the final exact answer is enclosed within oxed{} with latex format.
system
Two players face off in a dice combat game. The attacker will roll three dice and the defender will roll two. The attacker will defeat the defender in a single roll of dice if and only if both conditions below are satisfied:

i) The highest die of the attacker is greater than the highest die of the defender.

ii) The second highest die of the attacker is greater than the second highest die of the defender (we assume that the "second highest die" can be the same as the highest die if two or more dice tie for the highest value).

Given that all dice are fair and outcomes are equally probable, calculate the probability that the attacker wins when the defender rolls:

a) Two fives;

b) One five and one four.
user
\frac{43}{216}
rule
Two players face off in a dice combat game. The attacker will roll three dice and the defender will roll two. The attacker will defeat the defender in a single roll of dice if and only if both conditions below are satisfied:

i) The highest die of the attacker is greater than the highest die of the defender.

ii) The second highest die of the attacker is greater than the second highest die of the defender (we assume that the "second highest die" can be the same as the highest die if two or more dice tie for the highest value).

Given that all dice are fair and outcomes are equally probable, calculate the probability that the attacker wins when the defender rolls:

a) Two fives;

b) One five and one four.
train