+407 The dartboard in the diagram is composed of two concentric circles. The radius of the larger circle is twice as long as that of the smaller circle. The dartboard is further divided into four equal parts by two diameters of the larger circle. 200 darts are randomly thrown towards the dart board and 60% of them land off the dartboard. What is the expected number of darts that land in the blue region?
+36916 40 % of 200 hit the board = 80 darts
1/4 will hit in the upper R quadrant (20 darts)
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the area of the shaded region is 1/4 pi (2r)^2 - 1/4 pi r^2
4r^2 - r^2 = 3 r^2 out of total quadrant 1/4 pi (2r^2 )
3r^2 / 4r^2
3/4 of the 20 darts will hit the shaded area 3/4 x 20 = 15 darts ( as Cphil found below)
I saw Chris' answer and realized a mistake in my calcs....Thanx Chris !
+128475 Let the radius of the smaller circle = R and the radius of the larger circle = 2R
Then the total area = pi ( 2R)^2 = 4pi R^2
And the area of the outer circle pi [ ( 2R)^2 - R^2 ] = 3piR^2
And the blue area is 1/4 of this = (3/4)pi R^2
So....the probability of a dart landing in this area = (3/4)pi R^2 / ( 4 pi R^2) = 3/16
If 200 darts are thrown and 60% of them land off the dashboard, then 40% hit the dashboard = 80
So....the expected number of darts to land in the blue region =
(3/16) (80) = 15
