Two people roll a single six-sided fair die, each. $A$ rolls then $B$ rolls. The person with the higher number rolled wins (no one wins with a tie). What is the probability that $A$ wins? That $B$ wins? That neither wins?
I think I have to solve the problem using a system of equations and add up the probabilities at the end but I am not sure if that is the right idea. I just do not understand how to use this to solve the problem.
The probability that they tie is $\frac{1}{6}$ because no matter what the first person gets the second person has a $\frac{1}{6}$ chance of getting that number. The probability that $A$ wins is the same as the probability that $B$ wins. Therefore it is equal to $$\frac{1 - \frac{1}{6} }{2} = \frac{5}{12}$$