$X_1,X_2,...X_n$ and $Y_1,Y_2,..Y_n$ are independent random samples taken from the same continuous distribution with distribution function $F$.What is the probability that the inequality $Y_{(m')}<X_{(m+1)}<Y_{(m'+1)}$ where $0<m<n, 0<m'<n'$ will hold?
My approach: I wrote $P[Y_{(m')}<X_{(m+1)}<Y_{(m'+1)}]=\int P(Y_{(m')}<X_{(m+1)}<Y_{(m'+1)}|X_{(m+1)}=x)f_{(m+1)}(x) \text{dx}$
$ \implies \int P(Y_{(m')}<x<Y_{(m'+1)})f_{(m+1)}(x) \text{dx}$ Now I know the expression for the pdf $f_{m+1}(x)\\$. Is $P(Y_{(m')}<x<Y_{(m'+1)})={n \choose m'}(F(x))^{m'}(1-F(x))^{n-m'}?$ And is my approach correct?