Let $X,Y$ be the number of heads from two coins. What is $P(X=Y)$?

Question

You throw two fair coins $4$ times. Let $X$ be the number of heads from the first coin and let $Y$ be the number of heads from the second coin.

(a) What values can $X$ take? What is $\mathsf P(X = k)$?
(b) What is $\mathsf P(X = Y )$?
(c) What is $\mathsf P(X < Y )$?
(d) What is $\mathsf P(X > Y )$?

I know that $X$ can take values of $\{0,1,2,3,4\}$ as can $Y$, since both coin tosses are independent of each other. I also know that $$\mathsf P(X {=} k) ~=~\begin{cases} 0 & \textsf{ for all other $k$}\\1/16 & \textsf{ for }k \in\{ 0,4\}\\ 1/4 & \textsf{ for }k \in\{ 1,3\}\\3/8 & \textsf{ for }k = 2\end{cases}$$

What I am having trouble with is the $P(X = Y)$, but I think once I get help with that, parts c and d will make sense. However, wouldn't $P(X = Y)$ be the same values for $P(X = k)$? Please explain, and thank you!

Answer 1

By the law of total probability, we have that \begin{align*} P(X=Y) &= \sum_{k=0}^4 P(X=Y, Y = k) \\ &= \sum_{k=0}^4 P(X=k|Y=k)P(Y = k)\\ &=\sum_{k=0}^4 P(X=k)P(Y = k) \end{align*} where the last equality is true by independence.