You are participating in a not so popular game show You are

You are participating in a not so popular game show. You are in a room containing 2 identical big drawers, 100 red balls and 100 black balls. Any of the drawers is large enough to contain all the balls, and the balls are distinguishable only by their color. You are told that after you leave the room a blindfolded stranger will enter the room, open one of the drawers at random, and pick one of the balls inside that drawer at random (unless that drawer is empty, in which case he returns empty handed). You are instructed to distribute all the balls among the drawers, and told that you will win the prize if the stranger picks a red ball. Find the chance of winning for the following strategy: Place 30 red balls and 20 black balls in one dr awer and the remaining 70 red balls and 80 black balls in the other drawer. Show that if you choose a strategy for which one of the drawers (and hence both) has as many red balls as black balls, then your winning chance is 50% (assume none of the drawers was left empty). Find the best possible strategy and the chance of winning for that strategy.

Solution

a) probability of choosing any drawer is 1/2

if in first we place 30 red and 20 black the probability of gettinf red will be 30/50 = 0.6

the probability of winning will be 1/2*0.6 = 0.3

if we place 70 red and 0 black in 2nd drawer then proabability of red will be 70/150

probabaility of wiing will be 1/2*7/15= 0.233

so total chance of winning is 0.3+0.23 =0.53

b) if we choose a startegy in which the number of red balls in one drawer is equal to the number of black ball in that drawer then the number of red and black balls left will be equal and will be in the 2nd drawer.

example = if we choose to get 40 red and 40 black in 1st then by default rest 60 red and 60 black will be in 2nd drawer.

probability of red in 1st = 40/80 = 0.5

probability of winning = 1/2*0.5 = 0.25

probability of red in 2nd drawer = 60/120 =0.5

probability of winning = 1/2*0.5 = 0.25

adding both we will get 0.50

c) the best possible way is as given in part A