A three dice roll question I got this question from an interview.
A and B are playing a game of dice as follows. A throws two dice and B throws a single die. A wins if the maximum of the two numbers is greater than the throw of B. What is the probability for A to win the game?
My solution.
If $(X,Y)$ are the two random throws of A and $Z$ is the random throw of B, then the problem asks (if I guessed correctly) to compute $P(\max(X,Y)>Z)$. But how to do it? I highly appreciate your help.

To add to the other answers, while this isn't what you were exactly asking for, this is a good place to highlight that this type of problem is the exact type where numerical simulation is a powerful tool. A challenging mathematical problem is reduced to a very simple numerical problem. Here is some python code which simulates this game 1000 times and estimates the probability of A winning as the proportion of games A wins out of these 1000.
from numpy import random

def play_game(): 
    dieA_1 = random.choice([1,2,3,4,5,6])
    dieA_2 = random.choice([1,2,3,4,5,6])
    A_num = max(dieA_1, dieA_2)
    B_num = random.choice([1,2,3,4,5,6])
    
    A_wins = A_num > B_num 
    return A_wins

num_A_wins = 0 
for iter in range(1000): 
    A_wins = play_game()
    num_A_wins += A_wins  

print(f"probability of A winning is {num_A_wins/1000}")

Output:
probability of A winning is 0.582

