There are three caskets of treasure. The first casket contains 3 gold coins, the second casket contains 2 gold coins and 2 bronze coins, and the third casket contains 2 gold coins and 1 silver coin. You choose one casket at random and draw a coin from it. The probability that the coin you drew is gold has the value $\frac ab$, where $a$ and $b$ are coprime positive integers. What is the value of $a+b$?
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1The original MathCounts-type problem must have had "has the value $a/b$". Value $ab$ does not make sense, since the probability is clearly $\lt 1$. – 2012-11-12