When rolling a die, are we equally as likely of getting 1 twice as getting exactly a single 1 and a single 2?

Question

The die is unbiased and has $6$ sides. I want to know if the probability of getting $1$ twice (which is $(\frac{1}{6})^2$) is the same as getting a $1$ and a $2$ (order does not matter).

Answer 1

On the first roll, there is a $\frac{1}{6}$ chance that a one is rolled. On the second roll, there is also a $\frac{1}{6}$ that a one is rolled. Multiplying the probabilities, there is a $\frac{1}{36}$ chance that we roll two ones.

Now let us consider the other possibility. On the first roll, there is a $\frac{1}{3}$ chance that either a one or a two is rolled. No matter what happens, there is a $\frac{1}{6}$ chance we roll the number that was not rolled before (e.g. if one was rolled first, we rolled a two next). Multiplying the probabilities, there is a $\frac{1}{18}$ chance that we roll two ones.

Therefore, it is more likely we roll a two and a one than two ones.