I want to share a question that is really old (older than me, actually). It is from the June 1972 paper. As most students know, Maclaurin’s series was tested in last year A’levels Paper 1 as a sum to infinity. And this DRV did the same thing. Here is the question.
A bag contains 6 black balls and 2 white balls. Balls are drawn at random one at a time from the bag without replacement, and a white ball is drawn for the first time at R th draw.
(i) Tabulate the probability distribution for R.
(ii) Show that E( R ) = 3, and find Var( R ).
If each ball is replaced before another is drawn, show that in this case E( R ) = 4.
, if we do it with replacement.
![\mathbb{E}(R) = \frac{2}{8} [1 + (2) \times \frac{6}{8} + (3) \times (\frac{6}{8})^2 + (4) \times (\frac{6}{8})^3 + ... ]](http://theculture.sg/wp-content/ql-cache/quicklatex.com-b3a3b74af4a332c456bd3e422d759d8b_l3.png "Rendered by QuickLaTeX.com")
![\Rightarrow \mathbb{E}(R) = \frac{2}{8} [1 + (2) \times \frac{6}{8} + (3) \times (\frac{6}{8})^2 + (4) \times (\frac{6}{8})^3 + ... ]](http://theculture.sg/wp-content/ql-cache/quicklatex.com-aaaa257936eaeaccda3968f6d6d1adbb_l3.png "Rendered by QuickLaTeX.com")
