<s>[INST] probability problem and 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 $a+b$?
 [/INST] If we draw from the first, our probability of gold is $\dfrac{3}{3}$. If we draw from the second, our probability of gold is $\dfrac{2}{4}$. And if we draw from the third, our probability of gold is $\dfrac{2}{3}$. So the required probability is
$$\frac{1}{3}\cdot\frac{3}{3}+\frac{1}{3}\cdot \frac{2}{4}+\frac{1}{3}\cdot\frac{2}{3}.$$
This simplifies to $\dfrac{13}{18}$. The sum of numerator and denominator is $31$. 
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