How do you find missing probabilities from a distribution?

Question

I am given the following independent joint distributions:

\begin{align*} P(X=2, Y=0) &= 1/2\\ P(X=2, Y=1) &= 1/4\\ P(X=1, Y=0) &= p\\ P(X=1, Y=1) &= q \end{align*}

When solving for $p$ and $q$, I have determined that $\frac12 + \frac14 + p + q = 1$. Also, $p + q = \frac14$.

How do we further break this down to find the exact probabilities for $p$ and $q$?

Answer 1

You have $p + q = 1/4$. Since $X$ and $Y$ are independent you have $P(X = 1, Y = 0) = P(X = 1) P(Y = 0)$. Now $P(X = 1) = p + q = 1/4$ and $P(Y = 0) = 1/2 + p$. So you get the equation $p = (1/4)(1/2 + p)$, which you can solve for $p$.