Probability of rolling at least one of two specified numbers using 6-sided dice

Question

What is the formula - in simple language - to calculate the probability of rolling at least one of the two specified numbers using three 6-sided dice?

Let's say, the next roll of the three 6-sided dice needs to include at least one 5 or 6 (or both). In other words, I win if any of these show up at least once on any of the three dice on the next roll.

I do know the result should be 0.7037 but I could not reverse engineer it. There are 216 possible outcomes. I have a 0.33 probability of rolling either a 5 or a 6 with a single die, but I have three dice.

Answer 1

Our desired event is complementary to the event that none of the two specified number appears. Hence required probability is $$1-\frac{4^3}{6^3}=\frac{19}{27}\approx 0.7037$$

Answer 2

For questions like this (about "at least one...") a good trick is to compute your probability of failure.

For you to fail each die must come up 1,2,3, or 4. Since there's a $2/3$ probability of that happening for each individual die, the probability that it happens for all three of them is $(2/3)^3 = 8/27$. Thus, finally, the probability of you succeeding is $1-(2/3)^3 = 19/27 \approx 0.7037\dots$.