Let $X\_1,X\_2,....,X\_{n+1}$ independent uniform variables on the unit interval $[0,1]$, $V:=\max(X\_1,X\_2,...,X\_n)$ and $c \in [0,1].$ 

What is $\textrm{Prob}(X\_{n+1}<V|V<c)$?

Initially I thought this would be equivalent to 
$$\textrm{Prob}(X\_{n+1}<V<c)=\int\_0^cnz^ndz=\frac{n}{n+1}z^{n+1},$$

but I am concerned that equivalence may not hold.