1998 Paper 1 Q13

Year: 1998
Paper: 1
Question Number: 13

Course: LFM Stats And Pure
Section: Conditional Probability

Difficulty: 1484.0 Banger: 1532.0

conditional probability Bayes' theorem sampling with replacement combinatorial probability geometric series

Problem

I have a bag initially containing \(r\) red fruit pastilles (my favourites) and \(b\) fruit pastilles of other colours. From time to time I shake the bag thoroughly and remove a pastille at random. (It may be assumed that all pastilles have an equal chance of being selected.) If the pastille is red I eat it but otherwise I replace it in the bag. After \(n\) such drawings, I find that I have only eaten one pastille. Show that the probability that I ate it on my last drawing is \[\frac{(r+b-1)^{n-1}}{(r+b)^{n}-(r+b-1)^{n}}.\]

No solution available for this problem.

Rating Information

Difficulty Rating: 1484.0

Difficulty Comparisons: 1

Banger Rating: 1532.0

Banger Comparisons: 2

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Problem source

I have a bag initially containing $r$ red fruit pastilles
(my favourites)
and $b$ fruit pastilles of other colours. From time to time
I shake the bag thoroughly and remove a pastille at random.
(It may be assumed that all pastilles have an equal chance
of being selected.) If the pastille is red I eat it
but otherwise I replace it in the bag. After $n$ such
drawings, I find that I have only eaten one pastille.
Show that the probability that I ate it on my last drawing
is
\[\frac{(r+b-1)^{n-1}}{(r+b)^{n}-(r+b-1)^{n}}.\]

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