Again we had the solution to this question from Joel, well done Joel.
The solution is to put all the balls into one bag, then (for any number of balls, say n red and n white)
P(red) = 1/2 (1/2) + 1/2 (0) = 1/4.
The solution here is to put no white balls and at least one red in one bag, then the probability of choosing that bag and a red from it is 1/2, which is maximal. In order to make the probability of choosing a red from the other bag as large as possible we put as many reds in that bag as possible, which means putting exactly one red and no whites in the first bag and all the remaining balls in the second bag. With 4 red and 4 white balls the probability of selecting a red ball is then 5/7.
The case for n = 100 is left to the reader. In general, with 1 red in one bag and all the remaining balls in the other bag, the probability of drawing a red ball tends to a limit as n tends to infinity. You may like to find this limiting value.