Probability of 3 Sided Dice

Question

3 sided dice with the probability

$P(1) =0.5$

$P(2)=0.3$

$P(3)=0.2$

What is the probability of $(1)$ appear twice if I roll the dice $3$ times?

Here I use binomial

$\frac{3!}{2!} (0.5)^2 (0.5)$

Is that correct?

Answer 1

Yes, that is correct.

There are

$$3 = \frac{3!}{2!(3 - 2)!}$$

disjoint successful outcomes: $(1, 1, n)$, $(1, n, 1)$, and $(n, 1, 1)$. The probability of each is equal to

$$P(1)P(1)(1 - P(1)) = 0.5^2 \cdot 0.5$$

You add $3$ of these together, since any one of them is successful.