Question 255178: My homework is on geometric probability. One of my problems has to do with a dart board. I am given that the Diameter of the outer double ring = 41 cm. the diameter of the inner double ring = 36 cm. the diameter of the outer triple ring = 25.6 cm. the diameter of the inner triple ring = 21.2 cm. the diamer of the outer bulls eye = 4.4 cm. and finally the diameter of the inner bull's eye = 1.4 cm. My question asks What is A) what is the probability that the dart will land IN the double ring B) What is the probability of hitting the single bull's eye and finally C) what is the probability of hitting the triple 20 ring. Please help as much as possible, I'd really appreciate it. Thank you! ** Note: The numbers go from 1- 20 on a dart board if that helps at all.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! if we assume that you can hit anywhere in the scoring area of the dartboard 100% of the time, then the total scoring area of the dartboard represents 100% probability.
any area less than that on the dartboard means the probability of hitting that area = that area divided by the total area of the dartboard.
total area of the dartboard is equal to the area of the circle from the center of the circle to the outer double ring line.
the area of a circle is equal to pi*r^2 where r is the radius.
singe the diameter of a circle is equal to 2 times the radius of a circle, then the area of a circle can be shown as pi*(d/2)^2 where d is the diameter.
the scoring area of the dartboard is equal to pi * (41/2)^2 = pi * 20.5^2 = 1320.254313 cm^2.
if you look at the picture in the attached link, that's the area from the center of the dartboard to the double ring line.
that's the total scoring area of the dartboard and we are assuming you will hit in the scoring area of the dartboard 100% of the time.
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if you want to hit in the double scoring area, that's the area between the outer double ring line and the inner double ring line.
the diameter of the outer double ring line is 41 cm.
the diameter of the innter duble ring line is 36 cm.
the double ring area is the area from the center of the circle to the outer double ring line minus the area from the center of the circle to the inner double ring line.
the area between those lines is equal to pi*(41/2)^2 - pi*(36/2)^2 = pi*20.5^2 - pi*18^2 = 1320.254313 - 1017.87602 = 302.3782929 cm^2.
the probability of hitting in the double scoring area is therefore equal to:
302.3782929 cm^2. / 1320.254313 cm^2 = .229030339
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the probability of hitting the single bullseye depends on what you call the single bullseye.
you have an outer bullseye line and an inner bullseye line.
the very center is the double bullseye which means the single bullseye area must be between the outer bullseye line and the inner bullseye line.
this equals the area from the center of the circle to the outer bullseye line minus the area from the center of the circle to the inner bullseye line.
the area of the circle from the center to the outer bullseye line is equal to:
pi*(4.4/2)^2 = pi*2.2^2 = 15.20530844 cm^2.
the area of the circle from the center to the inner bullseye line is equal to:
pi*(1.4/2)^2 = pi*.7^2 = 1.5393804 cm^2.
the area between the outer bullseye line and the inner bullseye line is equal to 15.20530844 cm^2 - 1.5393804 cm^2 = 13.66592804 cm^2
the probability of hitting in the single bullseye area is therefore equal to:
13.66592804 cm^2 / 1320.254313 = .010350982
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the area of the triple 20 is equal to 1/20th of the triple area.
the triple area is the area between the outer triple line and the inner triple line.
the area from the center of the circle to the outer triple line is equal to:
pi * (25.6/2)^2 = pi * 12.8^2 = 514.7185404
the area from the center of the circle to the inner triple line is equal to:
pi * (21.2/2)^2 = pi * 10.6^2 = 352.9893506
the triple area is the area between these 2 lines which equals 514.7185404 minus 352.9893506 = 161.7291898 cm^2
the triple 20 area would be 1/20th of this which would equal 8.086459493
the probability of hitting into the triple 20 area is therefore equal to:
8.086459493 / 1320.254313 = .006124926
the whole thing depends on the folloowing:
*** you will hit the scoring area of the dartboard 100% of the time.
*** the probability of hitting one area of the scoring area of the dartboard is the same as the probability of hitting any other area of the scoring area of the dartboard.
*** the probability of hitting any area within the scoring area of the dartboard is based on the size of that area divided by the size of the scoring area of the dartboard.
any change in these assumptions changes the outcome.
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