Black and white balls probability

Question

A bag contains $5$ white balls. The following process is repeated. A ball is drawn uniformly at random from the bag (that is each of the five balls has equal probability $\frac{1}{5}$ of being drawn in each trial). If the color of the drawn ball is white then it is colored with black and put into the bag. If the color of the drawn ball is black then it is put into the bag without changing its color. What is the expected number of times the process has to be repeated so that the bag contains only black balls?

Now, I am new to probability and I thought of random brute force methods to work out this problem, but am unable to do so. I have no good ideas on how to do this problem. Any kind of help/hint is appreciated. Thank you.

Answer 1

Hint:

Use linearity of expectation. For any $5$ random variables $X_1,..,X_5$ whether we have independence or not

$$E(X_1+X_2+X_3+X_4+X_5)=E(X_1)+E(X_2)+E(X_3)+E(X_4)+E(X_5)$$

Answer 2

Let $X_k$ be event of drawing a new ball after $k$ balls were already drawn. The probability of the event is: $$ P(X_k)=\frac{n-k}{n}, $$ where $n$ is the number of balls.

By linearity of expectation we have: $$ E_n=E\left(\sum_{k=0}^{n-1}X_k\right)=\sum_{k=0}^{n-1} E(X_k) =\sum_{k=0}^{n-1}\frac{1}{P(X_k)}=\sum_{k=0}^{n-1}\frac{n}{n-k}=n\sum_{k=1}^n\frac{1}{k}. $$