Math Help - Two Urns Problem

1. Two Urns Problem

How can 20 balls, 10 white and 10 black, be put into two urns so as to maximize the probability of drawing a white ball if an urn is selected at random and a ball is drawn at random from it?

2. Hello, novas51791!

This is a classic (very old) puzzle.

How can 20 balls, 10 white and 10 black, be put into two urns
so as to maximize the probability of drawing a white ball
if an urn is selected at random and a ball is drawn at random from it?

Solution: . $\begin{array}{c}\text{Put 1 white ball in Urn A} \\ \text{put the rest in urn B}\end{array}$

We have: . $\boxed{\begin{array}{c} \text{Urn A} \\\text{1 W} \\ \\ \end{array}} \qquad \boxed{\begin{array}{c} \text{Urn B} \\\text{9 W} \\ \text{10 B} \end{array}}$

$P(\text{urn A}) \,=\,\frac{1}{2}$

. . $P(\text{W from urn A}) \,=\,1$
. . Hence: . $P(\text{urn A }\wedge\text{ W}) \:=\:\left(\frac{1}{2}\right)(1) \:=\:\frac{1}{2}$

$P(\text{Urn B}) \:=\:\frac{1}{2}$
. . $P(\text{W from urn B}) \:=\:\frac{9}{19}$
. . Hence: . $P(\text{urn B }\wedge\text{ W}) \:=\:\left(\frac{1}{2}\right)\left(\frac{9}{19}\ri ght) \:=\:\frac{9}{38}$

Therefore: . $P(\text{White}) \;=\;\frac{1}{2} + \frac{9}{38} \;=\;\frac{14}{19} \;\approx\; 73.7\%$

3. Thank you so much! That's the answer I got as well, from guessing and checking.

Is it just a matter of guessing and checking, or is there some mathematical way to obtain that answer? Thanks!