Maximizing the chances of picking up a red marble.

Question

You have 2 jars, 50 red marbles and 50 blue marbles. You have to place all the marbles into the jars such that when you blindly pick one marble out of one jar, you maximize the chances that it'll be red. You can arrange the marbles however you like, but each marble must be in a jar. There is equal probability of selecting a jar.

Answer 1

Start with 50 red marbles in the first bag and 50 blue marbles in the second bag. The probability of drawing a red marble in this case is $\frac{1}{2}$. If we move a red marble from the first bag to the second bag, the probability becomes \begin{equation*} \frac{1}{2}\left(1 + \frac{1}{51}\right) \end{equation*} So when we keep moving red marbles from the first bag to the second, the probability keeps increasing. The maximum is \begin{equation*} \frac{1}{2}\left(1 + \frac{49}{99}\right) \end{equation*} and is achieved when we have one red marble in the first bag and all other marbles in the second bag.