Let $T$ stand for a transplant, and $S$ stand for surviving 12 months or more.

$Pr[T|S]=\dfrac{Pr[S|T]Pr[T]}{Pr[S]}$

$Pr[S]=Pr[S|T]Pr[T]+Pr[S|!T]Pr[!T]$

$Pr[T]=0.38$

$Pr[!T]=1-Pr[T]=1-0.38=0.62$

$Pr[S]=(0.85)(0.38)+(0.30)(0.62)=0.509$

$Pr[T|S] = \dfrac{(0.85)(0.38)}{0.509}=0.634578$