The Traveller and Three Kegs

There are 3 kegs of water. The first keg has 1 liter, the second keg has 2 liters, and the third keg has 3 liters of water in it.

Each day our traveler picks one keg at random with each keg having the same chance of being picked, and drinks one liter of water from it. When he empties a keg, he throws it away. When he is left with one keg, he records the amount of water in it.

What is the expected amount of water (in liters) remaining in that last keg?

If this expected value can be expressed as ab \dfrac ab, where aa and bb are coprime positive integers, enter a+ba+b.


Note: You may use a calculator if you want.

Bonus: Solve this problem for 5 kegs.


Inspired by bobbym.
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