Question

There are two boxes, one containing 10 red balls and the other containing 10 green balls. You are allowed to move the balls between the boxes so that when you choose a box at random and a ball at random from the chosen box, the probability of getting a red ball is maximized. This maximum probability is:

Answer 1

the probability for selecting red box is 1/2 but randomly if we select green box, the probability will be 0.
so, total probability for this case is 0.5.

now consider the other case, 9 red balls are transferred to green ball box. now total balls in that box will be 19.

total probability = probability to select 1st box + probability to select 2nd box

= (1/2)*(1C1/1C1)+(1/2)*(9C1/19C1)

= (1/2)*(1+9/19)

= (1/2)*(28/19)

= 14/19