We have recently got a dart board for our house and I was wondering what the average score was if the dart randomly hit a scoring part of the board.

I’m not going to do it, but I can tell you how: work out the area of each scoring part (the measurments should be at an offical dart site) it’s easier to caculate the area of each circle (pi*r^2 obviously)on the dart board and then to get the area of each ring subtract the larger circle from the next smallest circlle, then to get the area of each individual scoring part just divide the area of the ring it is inn by 20. Then multiply each score by the area which it occupies, then divide the whole lot by 62 which will give you the total score

*last two words should be “average score”

You can simplify it a bit by observing that the average score of the “single” sections of the board is 10½. So the average of all the doubles is 21, and of all the trebles 31½. No need to divide each ring into 20 parts.

(And strictly speaking, since you’re only interested in the relative size of the 5 regions, you don’t need Pi either, just the squares of the distances from the centre of the board.)

Might as well answer the OP (making some guesses about the dimensions of a dart board):

inner bullseye (radius 0.5cm): area = 0.25 (times pi)

outer (0.5-1.5cm): area = 2.25 - 0.25 = 2

treble ring (10-11cm): area = 121 - 100 = 21

double ring (19-20cm): area = 400 - 361 = 39

entire scoring area (20cm): area = 400, hence remaining scoring area = 337.75.

So the average score is ( 10½ x 337.75 + 21 x 39 + 31½ x 21 + 25 x 2 + 50 x 0.25 ) / 400 = 12.72.