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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?

 May 29, 2021
 #1
avatar+35066 
+3

40 % of 200 hit the board = 80 darts

 

1/4 will hit in the upper R quadrant  (20 darts)

 

**** edited ***

 

   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 !

 May 29, 2021
edited by ElectricPavlov  May 29, 2021
 #2
avatar+121063 
+3

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

 

 

cool cool cool

 May 29, 2021

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