Sample space in probability

Question

If the sample space is all possible rolls of two dice, how many outcomes have at least one four?

I think this to be $\frac{1}{4}$, but I think I am wrong. What is a correct way to calculate this?

Answer 1

You can just count them. If the first die equals 4, the other die can equal any value. If the second die equals 4, the first die can equal any value. However, we now counted (4, 4) twice, so the total number of possibilities equals:

$$2 \cdot 6 - 1 = 11$$

Since there are $6^2 = 36$ possible outcomes, the probability of rolling at least one 4 equals $\frac{11}{36}$.

Answer 2

Having at least one four is the complementary of having no fours, so $1-(\frac{5}{6})^2 = 30.\bar 5 \% = \frac{11}{36}$ of the $36$ possible draws, that is $11$.

Answer 3

If the sample space is all possibilities of two six-sided dice, then there are a total of $6^2$ combinations. Then, you want to know how many outcomes have at least a single $4$ in them? Well, we could have: $$\left\{ (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (1,4), (2,4), (3,4), (5,4), (6,4) \right\}$$ where the ordered pairs reflect the roll of the first dice and second dice respectively. So that the probability would be: $$\dfrac{11}{36}$$