A four sided, balanced die has the values 1- 4. There are three successive throws. What is the probability that exactly two throws are the same value?

Question

anonymous · Answer

For questions involving "exact" probabilities, you'll often be able to assume a binomial distribution. Have you learned about it yet?

anonymous · Answer

No, I don't know what that is

anonymous · Answer

Alright, it's not needed for this problem (just another way of getting the same result).

The probability you want is given by $(.25)(.25)(.75)$. The reason why is because two of the rolls must yield the same value, and the third is anything else.

anonymous · Answer

Okay, I see.

anonymous · Answer

Wait, I'm missing something. It should be $\color{red}3(.25)(.25)(.75)$.

anonymous · Answer

I see, since the dice is thrown 3 times

anonymous · Answer

Because there are 3 ways of getting 2 of the same dice side and 1 different from the others.

Suppose $n$ is the side you want and $n'$ is any side you don't want. You can have
$$n~n~n'\
n~n'~n\
n'~n~n$$

anonymous · Answer

No, it has to do with the order in which the sides come up. There are three ways you can have an extraneous dice roll.

anonymous · Answer

Okay, that makes sense now

anonymous · Answer

So the fact that there are 3 successive throws, doesn't matter? Or does that fit in somehow?