Probability Problem: Red and Blue Marbles in Jars

You have two jars, 50 red marbles and 50 blue marbles. You need to place all the marbles into the jars such that when you blindly pick one marble out of one jar, you maximize the chances that it will be red. When picking, you’ll first randomly pick a jar, and then randomly pick a marble out of that jar. You can arrange the marbles however you like, but each marble must be in a jar.

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Answer:

Let us assume that we put all the red marbles in jar A and all blue marbles in jar B. Then the probability of getting a red marble is:

jar A : (1/2)*1 = 1/2 (selecting the jar A = 1/2, red marble from jar A = 50/50)

jar B : (1/2)*0 = 0 (selecting the jar B = 1/2, red marble from jar B = o/50)

So probability of getting red marble is 1/2 . Now as we need to maximize the P (getting a red marble), we have to increase the probability of getting a red marble in jar B. If we select jar A, then getting a red marble is guaranteed, but it will also be guaranteed if there is only one red marble in that jar, then also the probability of getting a red marble from jar A is 1/1=1. So now we can place remaining 49 red marbles in jar B, so it increases the probability of getting red marbles in jar B.

So the maximum probability will be :

jar A : (1/2)*1 = 1/2 (selecting the jar A = 1/2, red marble from jar A = 1/1)

jar B : (1/2)*(49/99) = 0 (selecting the jar B = 1/2, red marble from jar B = 49/99)

Total probability = 74/99 (~3/4)