The maximum probability is 0.5.
To maximize the probability of picking a red ball after randomly selecting a bag and then randomly selecting a ball, we need to concentrate the red balls in one bag and the blue balls in the other. This way, when we randomly select a bag, the bag with more red balls is more likely to be chosen, increasing the overall probability of selecting a red ball.
Since there are 50 red balls and 50 blue balls, we can place all 50 red balls in one bag and all 50 blue balls in the other bag. This arrangement ensures that the bag containing all red balls has the highest probability of being selected, and when a ball is randomly selected from this bag, it is guaranteed to be red.
Therefore, the maximum probability of picking a red ball is 1, as we have ensured that the bag with red balls is always selected, and the ball picked from this bag is always red.
Question:
You have 50 red balls, 50 blue balls and 2 bags. You want to arrange the balls into these bags in such a manner as to maximize the probability that picking a bag at random and then picking a ball at random results in a red ball. What is the maximum probability?