Probability of Multiple Random Variables

I studying for the final of my probability course. I am coping with a question. I cannot figure out any logic to answer it.

Let X be the number of questions asked to you and let Y be the number of questions you answered wrongly.

The probability that you answer a question independently wrong is 1/4.

You are asked 0, 1 or 2 questions.

I have a problem with the probability of X = 0 and Y = 0(Y = 1 and Y = 2 as well).

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the question seems pretty confusing. you can try to make it a bit clearer. – mathemagician Jan 7 '13 at 13:54
Perhaps if you did not try to put into a nutshell, but rather used the edit button below your post and just typed in the exact statement of the question, it would help people understand what you are asking. If you will be translating the question into English, please be extra careful with the meanings of words. – Dilip Sarwate Jan 7 '13 at 13:57

Hint: The random variable $(X,Y)$ can take on values $(0,0), (1,0), (1,1), (2,0), (2,1), (2,2)$ and you are told that the conditional distribution of $Y$ given $X = x$ is a binomial distribution with parameters $\left(x, \frac{1}{4}\right)$. To derive the joint distribution (and hence $P\{(X,Y) = (0,0)\}$), you need the distribution of $X$, which is not specified in your revised problem statement. Perhaps you did not include this, or you are expected to assume that $X$ is equally likely to take on values $0,1,2$, or $X$ itself has a binomial distribution with parameters $(2,p)$ for some $p$.