|
|
STEP III 2007 question 14 solutionTSR 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) = (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. |