A die is thrown twice. What is the probability that at least one of the two dice shows the number 4?

In cases of question asking the probability of an event happening at least one time, it is usually easier to calculating the probability of the event not happening at all and subtracting it from 1.
In this case, we can find the probability of both dices rolling any number but 4, so each dice will have 5 available options. 
The probability of this happening on one dice is $$\frac{5}{6}$$ and the probability of this happening on both the dices simultaneously is $$\frac{5}{6}\times\frac{5}{6}=\frac{25}{36}$$

Now subtracting this from 1 gives us the required probability, $$1-\frac{25}{36}=\frac{11}{36}$$

Hence, Option B is the correct answer.