Interesting Original Probability Question

Question

I have 100 balls, which are all initially yellow. Every minute, I randomly choose a ball and paint it red.

How many balls are expected to be red after 100 minutes? Note: I could pick up a ball that's already painted red.

Answer 1

Consider a random variable $X_i$ such that $X_i = 1$ if the $i^{th}$ ball is red after 100 minutes. Note that $Pr(X_i = 1) = 1 - ({99 \over 100})^{100}$. Using linearity of expectation,

$$E(X) = \sum_{i=1}^{100} E(X_i) = 100\left(1 - \left({99 \over 100}\right)^{100}\right) \approx 63.4$$