Dartboard Probability

Probability Level 2

The dartboard above is made up of three concentric circles with radii 1 , 3 , 1, 3, and 5. 5. Assuming that a dart thrown will land randomly on the dartboard, what is the probability that it lands in the green region?

32% 36% 40% 60%

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Eli Ross by Eli Ross

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7 solutions

Eli Ross Staff
Sep 18, 2015

The area of the green circle, ignoring the red circle, is 3 2 π = 9 π 3^2 \pi = 9\pi . The area of the red circle is 1 2 π = π . 1^2\pi = \pi. Thus, the area of the green region is 9 π − π = 8 π . 9\pi - \pi = 8\pi.

The area of the entire dartboard is 5 2 π = 25 π , 5^2\pi = 25\pi, so the probability that it is in the green region is 8 π 25 π = 32 % . \frac{8\pi}{25\pi} = 32\%.

56 Helpful 1 Interesting 6 Brilliant 2 Confused

With the potential for paddling, I believe the possibilities must be carefully estimated further, because sound scare, water upset, etc. must be included. I don't think simple mathematics is all that is needed.

Robert Fenhagen - 4 years, 9 months ago
Rajendra Patil
Sep 20, 2015

32% .The areas of the three circles are 1, 9 and 25 pie squares. The area where the dart can hit will be 1,8,16 pie squares. For the dart to hit the green area the probability will be 8/25 ie 32%.

8 Helpful 0 Interesting 0 Brilliant 0 Confused

Says I am incorrect and I had the same reasoning

Sea Shell - 3 years, 3 months ago
Tridip Das
Sep 20, 2015

The area of the red, green and blue shaded regions are π,8π,16π, So probability that the dart will hit the green region is =8π/(π+8π+16π) Which is equal to 8/25or 32%

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Varun Parkash
Sep 21, 2015

The explanations that people have given, i think, are not correct. If the "circles" are of radii 1, 3, and 5, then the regions will have the areas - π, 3π and 5π respectively (using ring formula where the area of the ring = π ( R 2 R^{2} - r 2 r^{2} ) The answer will be the same, but explanation different.

1 Helpful 0 Interesting 0 Brilliant 1 Confused

I was thinking with that in mind but slightly different, π(r^2)(2/5) = 25π (2/5) = 10π or 40% where did I go wrong?

Joel S - 3 years, 1 month ago
Brian Falcasantos
Sep 20, 2015

My answer is far less accurate and probably involves some luck. I figured out the area of each circle and then subtracted the area of the inner circle as shown in Column D. I then divided that number (25.12) by the sum of Column D (75.36) = .3333 which was closer to 32% than it was to 36%.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

That is the correct approach. The reason why your answer is off slightly, is because you used approximate values to find n π n\pi , instead of just leaving it in terms of π \pi .

That is the correct approach. The reason why your answer is off slightly, is because you used approximate values to find n π n\pi , instead of just leaving it in terms of π \pi .

Calvin Lin Staff - 5 years, 8 months ago

All 3 regions together is 100%. That is an area of 25 pi. So, each pi represents an area of 4. The ring in the middle has an area of 8 pi. And, 8 x 4 = 32%.

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Cade Bryant
Nov 3, 2016

The easiest way is to factor out π \pi after calculating the areas of the circles (since they all involve it) and just deal with the coefficients:

r = 1 g = 8 (i.e., 9 - r) b = 16 (i.e., 15 - g - r)

Thus, ratio of g to entire set of circles is 8/(16+8+1) = 0.32.

0 Helpful 0 Interesting 0 Brilliant 1 Confused

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