Red and Blue Marbles
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.
Given that the number of red and blue marbles are the same, we would tend to think that the odds are 50-50. we would try different combinations, such as 25 of each colored marble in a jar or putting all red marbles in one jar and all the blue in the other. we will still end up with a chance of 50%.
So lets think of a better way to distribute the marbles. What if we put a single red marble in one jar and the rest of the marbles in the other jar? This way, we are guaranteed at least a 50% chance of getting a red marble (since one marble picked at random, doesn’t leave any room for choice). Now that you have 49 red marbles left in the other jar, you have a nearly even chance of picking a red marble (49 out of 99).
So let’s calculate the total probability.
P( red marble ) = P( Jar 1 ) * P( red marble in Jar 1 ) + P( Jar 2 ) * P( red marble in Jar 2 )
P( red marble ) = 0.5 * 1 + 0.5 * 49/99
P( red marble ) = 0.7474
Thus, we end up with ~75% chance of picking a red marble.
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