First Repeated Die Roll

You roll a fair six-sided die repeatedly until you see a number that has already appeared. Let $r$ be the total number of rolls, and let $p_r$ be the probability that the game lasts exactly $r$ rolls. **Part 1.** Calculate $p_3$. **Part 2.** What is $p_1 + p_2 + \cdots + p_{10}$? **Part 3.** What is $P(r = 5 \mid r \geq 3)$?