You have two jars, 50 red marbles and 50 blue marbles. You need 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 will be red. When picking, you will first randomly pick a jar, and then randomly pick a marble out of that jar. You can arrange the marbles however you like, but place all 100 marbles in the jars.

Probability of selecting any one jar is 1/2. Now we want to place these 100 marbles in 2 jars to maximize the probability of red marble pick. We can put 1 red marble in one jar and rest all the 99 marbles (50 blue coloured and 49 red coloured) in other jar.

Now, calculating the probability for red marble pick in the first chance goes this way:

In the first jar in which there is only 1 red marble and the probability of picking red is:

(1/2) * (1/1) = 1/2

If the second jar is considered then it's: (1/2) * (49/99) = 49/198.

So, for calculating the total probability considering both the jars can be done in following way:

((1/2) * (1/1)) + ((1/2) * (49/99)) = (1/2) + (49/198) = 148/198 = 74/99 = 0.7474

That is 74.74 %.

nice - what if you want to maximize drawing of n consecutive reds ? --> n reds in first jar, rest in other ??