He should place ONE white marble in bowl A and the rest in bowl B.

. . . . . $\displaystyle \boxed{\begin{array}{c}\text{Bowl A} \\ \text{1 white} \\ \\ \end{array}} \qquad \boxed{\begin{array}{c}\text{Bowl B} \\ \text{49 white} \\ \text{50 black} \end{array}} $

$\displaystyle P(\text{bowl A}) \:=\:\frac{1}{2},\quad P(\text{white}) \:=\:1$

. . Hence: .$\displaystyle P(\text{white from bowl A}) \;=\;\frac{1}{2}\cdot1 \:=\:\frac{1}{2}$

$\displaystyle P(\text{bowl B}) \:=\:\frac{1}{2},\quad P(\text{white}) \:=\:\frac{49}{99}$

. . Hence: .$\displaystyle P(\text{white from bowl B}) \:=\:\frac{1}{2}\cdot\frac{49}{99}$

Therefore: .$\displaystyle P(\text{white}) \;=\;\frac{1}{2} + \frac{49}{198} \;=\;\frac{74}{99}$