The standard international dartboard is known as the 18-inches-in-diameter “cloc

Question

The standard international dartboard is known as the 18-inches-in-diameter “clock” board. Whenever a dart hits the bull’s-eye (means that a dart falls within 1 inch of the center), 50 points are scored. If the point of impact is between 1 and 3 inches from the center, 30 points are scored, if it is at a distance of 3 to 5 inches, 20 points are scored and if it is at a distance of more than 5 inches, 10 points are scored. Assume that Sam is skilled enough to be able to throw a dart within the boundaries of the board. Sam can place the dart uniformly on the board (i.e., the probability of the dart falling in a given region is proportional to its area and the area of a circle is r2 where r is the radius of a circle).

Sam is obsessed with getting the highest point. Determine,

i) On average, how many darts does Sam need to throw in order to hit the bull’s-eye?

ii) On average, how many darts does Sam need to throw in order to hit the bull’s-eye five times?

Explanation / Answer

probabilty of hitting the bull's eye =(d/D)2  ; where d is dia of bulls eye and D is diameter of Board

=(2/18)2 =1/81

i) hence from geomteric mean number of darts Sam need to throw in order to hit the bull’s-eye=1/p=81

(ii)number of darts Sam need to throw in order to hit the bull’s-eye five times =r/p=5*81=405

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