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 have equal probability (= 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 (up to two decimal places) the process has to be repeated so that the bag contains only black balls.


Define $X_i$ Such that $X_i$ denote the number of trials required to observe the $i^{th}$ white ball after observing $i-1$ white balls.

So the expected number of times the process has to be repeated is $E[\sum_{i=1}^{5}{X_i}] = \sum_{i=1}^{5}{E[X_i]}$. Note that $X_i$ follows geometric (number of trails for first success) with probability $\frac{6-i}{5}$ hence $E[X_i] = \frac{5}{6-i}$ so just sum this up from 1 to 5 to get the answer.

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    $\begingroup$ @MartijnWeterings I actually made a mistake in defining $X_i$ properly...now it is correct, please have a look? $\endgroup$ – Vishaal Sudarsan Mar 13 '18 at 7:52
  • $\begingroup$ @MartijnWeterings It is not an indicator variable. I made a mistake writing it, it clearly follows geometric. $\endgroup$ – Vishaal Sudarsan Mar 13 '18 at 8:02
  • $\begingroup$ @MartijnWeterings and what is the difference? ...It is the same as the number of trails requierd to observe the ith white ball. $\endgroup$ – Vishaal Sudarsan Mar 13 '18 at 10:24
  • $\begingroup$ Let us continue this discussion in chat. $\endgroup$ – Vishaal Sudarsan Mar 13 '18 at 10:34