Fifty red and fifty blue marbles probability puzzle

You have two jars and hundred marbles.

Fifty of the marbles are red, and fifty are blue.

You must place all 100 marbles in the jars.

You can decide how many and what marbles to be placed in each of the jars.
Each jar can hold 100 marbles.
One of the jars will be chosen at random; then 1 marble will be withdrawn from that jar at random.

How do you maximize the chance that a red marble will be chosen?

What is the chance of selecting a red marble when using your scheme?

Solution:

At first glance, it looks impossible to stack the odds one way or another.

The number of red and blue marbles is exactly equal.

You have to use them all; you cannot "lose" a few blue marbles.

The way a marble is chosen is entirely "random."

Shouldn't the chance of drawing a red marble be fifty-fifty?

It is when you put 25 marbles of each color in each jar.

In fact, the odds are fifty-fifty when there are 50 marbles in each jar, regardless of how the colors are mixed.

Put all the red marbles in jar A and all the blue marbles in jar B.

Then the chance of drawing a red marble is still exactly 50 percent, for that is the chance that jar A is chosen (guaranteeing that the marble selected at random from it will be red). This suggests the puzzle's answer.

You really don't need all 50 red marbles in jar A. One marble would do just as well.

In that case, there is still a 50 percent chance that jar A, containing a lone red marble, will be chosen.

Then the red marble will be "chosen" at random - not that there is any choice.

That yields a 50 percent chance of choosing a red marble just for jar A.

You still have 49 more red marbles, which you can and must put in jar B.

In the event jar B is chosen, you have nearly an even chance of drawing a red marble.

Actually the chance is forty-nine in ninety-nine.

The total chance of selecting a red marble with this scheme is just under 75 percent

(50 percent + 1/2 of 49/99, which comes to about 74.74 percent).

Source:
"How Would You Move Mount Fuji?"

Microsoft's Cult Of The Puzzle
By William Poundstone