A cookie jar contains 10 cookies of 3 types. There are 5 chocolate-chip cookies, 3 oatmeal-raisin,cookies, and 2 sugar cookies. You reach into the jar and choose a cookie at random and then, without replacing the first cookie, reach into the jar and choose another cookie at random. What is the probability that both of the cookies you choose are the same type?

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We should find the probability of obtaining 2 of each cookie.

$$ P(\text{two chocolate}) = \frac{5}{10}\cdot \frac{4}{9} = \frac{20}{90}$$ $$ P(\text{two oatmeal-raisin}) = \frac{3}{10}\cdot \frac{2}{9} = \frac{6}{90} $$ $$ P(\text{two sugar}) = \frac{2}{10}\cdot\frac{1}{9} = \frac{2}{90} $$

Summing them up to get the probability of obtaining the same type:

$$ P(\text{same type}) = \frac{20}{90} + \frac{6}{90} + \frac{2}{90} = \boxed{\frac{28}{90}} $$