Ranson holds $3$ white and $9$ purple jelly beans in his hand. Nancy holds $2$ white, $5$ yellow, and $2$ purple jelly beans in her hand. Each randomly picks a jelly bean to show the other. What is the probability that the colors match? (adapted from $2013$ AMC $8$ Problem, Question \# $14$) 
Overseas Competition->Knowledge Point->Counting Modules->Statistics and Probability->Questions Involving Probability->Basic Concepts of Probability
The probability that both show a white bean is $\dfrac{3}{12}\times \dfrac{2}{9}=\dfrac{1}{18}$. The probability that both show a purple bean is $\dfrac{9}{12}\times \dfrac{2}{9}=\dfrac{1}{6}$. Therefore, the probability is $\dfrac{1}{18}+\dfrac{1}{6}=\frac29$. 