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.
The probability distribution is as follows
Observe that the probability of a white ball is being drawn is always , if we do it with replacement.
And this will go on…
From MF26, we observe that
Then we have that