We have two random variables $X$ and $Y$ both of them follows Binomial Distribution with parameters $n$ and $0.5$. Consider, another random variables, $Z$ which is also Binomial Distribution with parameters $2n$ and $0.5$.
Then, represent the probability of $P(X=Y)$ in terms of probability of $Z$.
My approach
Clearly, the random variable, $Z$ is the sum of $X$ and $Y$ random variables. So,
$P(X=Y) = P(X-Y=0)$ which can only happen when both $X$ and $Y$ is zero. Hence,
$P(X-Y=0)=P(X=0)P(Y=0)=(\frac{1}{2})^n(\frac{1}{2})^n =(\frac{1}{2})^(2n) =P(Z=0)$
But, I am not sure I have done it correctly or not. The answer in the text is $P(Z=n)$
Any help.?