50 red marbles and 50 blue marbles Puzzle

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You have two jars, 50 red marbles and 50 blue marbles.
A jar will be picked at random, and then a marble will be picked from the jar. Placing all of the marbles in the jars.
How can you maximize the chances of a red marble being picked?

What are the exact odds of getting a red marble using your scheme?

SherlockHolmes Expert Asked on 29th September 2015 in

Seems tricky at first, right ?  Given that the number of red and blue marbles are the same, you would tend to think that the odds are 50-50. You 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. You would still end up with a chance of 50%.

So lets think of a better way to distribute the marbles. What if you put a single red marble in one jar and the rest of the marbles in the other jar? This way, you 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.

NitinGurbani Genius Answered on 2nd October 2015.

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