# Finding probabiliy mass function of two coin tosses

Consider an experiment of tossing two coins, denoted by $a$ and $b$, three times. Coin $a$ is fair (i.e., the probability of the head or the tail is $1/2$), while the coin $b$ is not and its probability of the head is $1/4$ and the tail is $3/4$. Let $X$ be the number of heads resulting in the three tosses of the coin $a$, and let $Y$ be the number of heads resulting in the three tosses of the coin $b$. Consider bivariate r.v. $(X; Y )$.

1 -Find the pmf of $(X,Y)$.

Progress: I have found the probabilities of $X$ and $Y$ separately meaning I have $P(x=0)$,$P(x=1)$, $P(x=2)$, $P(x=3)$ and similarly for $Y$. Now I don't know how to put them all together.

2- Find $P(X > Y)$

With $\mathbb{P}[X=i]$ and $\mathbb{P}[Y=i]$ figured out for $i\in\{0,1,2,3\}$ you are done, since $\mathbb{P}[X=i,Y=j]=\mathbb{P}[X=i]\mathbb{P}[Y=j]$ for any $i\in\{0,1,2,3\}$ and $j\in\{0,1,2,3\}$ since the two coin tosses are independent.
$\mathbb{P}[X>Y]$ is just sum of $\mathbb{P}[X=i,Y=j]$ over $i\in\{0,1,2,3\}$ and $j\in\{0,1,2,3\}$ such that $i>j$.