1998 Paper 1 Q14

Year: 1998
Paper: 1
Question Number: 14

Course: LFM Stats And Pure
Section: Geometric Probability

Difficulty: 1500.0 Banger: 1498.5

geometric probability circular coin grid lines optimisation expected value area ratio intersection coverage

Problem

To celebrate the opening of the financial year the finance minister of Genland flings a Slihing, a circular coin of radius \(a\) cm, where \(0 < a < 1\), onto a large board divided into squares by two sets of parallel lines 2 cm apart. If the coin does not cross any line, or if the coin covers an intersection, the tax on yaks remains unchanged. Otherwise the tax is doubled. Show that, in order to raise most tax, the value of \(a\) should be \[\left(1+{\displaystyle \frac{\pi}{4}}\right)^{-1}.\] If, indeed, \(a=\left(1+{\displaystyle \frac{\pi}{4}}\right)^{-1}\) and the tax on yaks is 1 Slihing per yak this year, show that its expected value after \(n\) years will have passed is \[ \left(\frac{8+\pi}{4+\pi}\right)^{n}.\]

No solution available for this problem.

Rating Information

Difficulty Rating: 1500.0

Difficulty Comparisons: 0

Banger Rating: 1498.5

Banger Comparisons: 2

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Problem source

To celebrate the opening of the financial year
the finance minister of Genland flings a Slihing, a circular
coin of radius $a$ cm, where $0 < a < 1$, onto a 
large board divided
into squares by two sets
of parallel lines 2 cm apart. If the coin does not
cross any line, or if the coin covers an intersection,
the tax on yaks remains unchanged. Otherwise
the tax is doubled. Show that, in order to raise most tax,
the value of $a$ should be
 \[\left(1+{\displaystyle \frac{\pi}{4}}\right)^{-1}.\]
If, indeed, $a=\left(1+{\displaystyle \frac{\pi}{4}}\right)^{-1}$
and 
the tax on yaks is 1 Slihing per yak this year, show that
its expected value after $n$ years will have passed is
\[ \left(\frac{8+\pi}{4+\pi}\right)^{n}.\]

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