Box 1 contains 5 red and 3 blue marbles while box 2 contains 2 red, 3 blue, and 1 white marble.
If you select a box at random and select a marble at random from it, then toss that marble into the other box and randomly draw a marble from it, what is the probability that the marble you draw is blue?
1. Approximately 43.55 percent
4. Approximately 10.71 percent
Events "tossing marble from Box1 into Box2" and "tossing marble from Box2 into Box1" are equally probable. What matters in both cases is the color of the marble being tossed into the second box, and that depends on the box from which it originates.
Numbers in the first summand are derived as follows:
(probability that you have the case of Box1->Box2 marble tossing)*((probability that the blue marble is selected from box1 and tossed into box2)*(probability that the blue marble is drawn from box2 now that you have one more blue marble in it)+(probability that the marble from box1 tossed into box2 is of some color other than blue)*(probability that the blue marble is drawn from box2 now that you have one more marble in it of some color other than blue))
Second summand covers the case box2->box1
Scenarios box1->box2 and box2->box1 are equally likely since you make a decision at random. When choosing among n possibilities at random, each possibility is equally probable with probability 1/n. That is what making a random decision means, 'its all the same to you', or just word 'whatever'.