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STEP III 2007 question 14 solution
From The Student RoomTSR Wiki > Study Help > Subjects and Revision > Mathematics > STEP > STEP III 2007 question 14 solution (i) Consider the circle C of radius R (<=1) at the origin. The chance of a dart hitting C is In the case of the n-1 nearest darts, the chance of n-1 darts lying within a radius R is p(n darts lie within) + p(n-1 darts lie within) = Then the expected area is (ii) Now define S to be the square with sides 2M centered at the origin. The chance of a dart hitting S is (iii) Clearly greater, since the alllowed 'target' for the square dartboard includes the entirety of the 'target' for the circular one. For each dart, there's a chance Comment: At first, I took (iii) to be exclusively considering the square dartboard and comparing the area of the bounding circle to the area of the bounding square (which is a complete misread of the question - doh!). But it's quite an interesting question: it's clear the square is better for large n (since you will typically end up with at least one point near the corners of the dartboard, and bounding the corners gives a square of area 4 but a circle of area Solution by DFranklin. |
. So the chance of n darts all hitting C is 
. But this is exactly the chance of the maximum radius r being 
. So 
. So r has p.d.f. 
. Then the expected area is 
.
= 
. Again differentiate to get the p.d.f. 
.
![\displaystyle 2n(n-1)\pi\int_0^1 (R^{2n-3} -R^{2n-1})R^2 \text{ d} R = n(n-1)\pi \left[\frac{1}{n}-\frac{1}{n+1}\right] = \frac{n-1}{n+1}\pi](http://thestudentroom.co.uk/../latexrender/pictures/74e3d277dd4c95ee18efafce742c37be.png "\displaystyle 2n(n-1)\pi\int_0^1 (R^{2n-3} -R^{2n-1})R^2 \text{ d} R = n(n-1)\pi \left[\frac{1}{n}-\frac{1}{n+1}\right] = \frac{n-1}{n+1}\pi")
.
, so as in (i) we find the minimum square side m has p.d.f. 
. Then the expected area is 
.
of hitting the target in (i) with the same expectation, and a chance 
of going outside the target in (i) with a resultant expected area > 
(which is greater than the expected area in (i)), so the expectation for the area must be greater than in (i). Similarly for considering [i]n[/i] darts.
), and simulation (or calculus) shows it's also better for small n. But I don't see a way of doing this without calculation. (Of course, there may not be one - it was only misreading the question that had me trying to find one after all!).
