If I roll a fair, regular six-sided die four times, what is the probability that I will roll the number exactly three times?

We have a (1/6)  probability of rolling any number on any roll.....so...to roll the same number three times out of four we have

C(4,3)  (1/6)^3 (5/6)  =  5/324  ≈ 1.5%