Variation of coupon collecting problem

Question

I have a doubt in coupon collecting problem. I am not able to understand how to approach this problem. "Suppose you already have 10 different coupons when there are 20 coupon types. What is the expected number of boxes for obtaining a coupon different from the 10 you already have?"

Answer 1

Hint: The probability of getting a box with a different coupon from the $10$ already gotten would be $\frac12$. The expected value for the number of trials to get a result that has probability $p$ is $$ \begin{align} \sum_{k=1}^\infty kp(1-p)^{k-1} &=p\frac1{(1-(1-p))^2}\\ &=\frac1p \end{align} $$