Two identical dice are thrown simultaneously. Find probability of getting a $3$ and a $2$.

Question

Two identical dice are thrown simultaneously. Find the probability of getting a $3$ on one of the dice while a $2$ on the other.

I'm pretty sure the answer is $\frac{2}{36}$ but my friend says the answer should be $\frac{1}{21}$.

He argues that as the dice are identical, so sample space comprises only $21$ combinations.

I'm unable to explain it to him but I don't agree with his answer of $\frac{1}{21}$.

Can someone tell me the correct answer and provide a short explanation?

Answer 1

There might be 21 possible outcomes ( (1,1), (1,2),....(6,6) ) , but they don't have equal probability, 1+1 can only occur in one way, but 2+3 can occur two different ways. There are 6 ways to throw (1,1), (2,2), (3,3), etc, and 15 different pairs (1,2)/(2,1), (1,3)/(3,1), etc, making 36 possible combinations. We can get a 2 and 3 in two different ways, so the probability is 2/36 as you say.

Answer 2

There are 36 possible outcomes. If it doesn't matter the order of the die, then the probability is $2* (1/36)$ ways it can happen. If it has to be 2 first, then 3 next, then it is $1/36$.