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A game at the state fair has a circular target with a radius

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by BTGmoderatorLU » Tue Oct 30, 2018 2:43 pm

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Source: Economist GMAT

A game at the state fair has a circular target with a radius of 10 cm on a square board measuring 30 cm on a side. Players win prizes if they throw two darts and hit only the circular area on both throws. If Jim hit the board on both throws, what is the probability that Jim won the game?

A. 1/3
B. 1/9
C. π/9
D. π^2/81
E. π^2/9

The OA is D.
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by fskilnik@GMATH » Tue Oct 30, 2018 5:28 pm
BTGmoderatorLU wrote:Source: Economist GMAT

A game at the state fair has a circular target with a radius of 10 cm on a square board measuring 30 cm on a side. Players win prizes if they throw two darts and hit only the circular area on both throws. If Jim hit the board in random points on both throws, what is the probability that Jim won the game?

A. 1/3
B. 1/9
C. π/9
D. π^2/81
E. π^2/9
$$? = P\left( {2\,\,{\rm{hits}}\,\,{\rm{in}}\,\,{\rm{circular}}\,\,{\rm{region}}\,\,{\rm{in}}\,\,{\rm{two}}\,\,{\rm{throws}}} \right)$$
We may consider the similar figure(!) in which the radius of the circular target is 1cm and the square board has 3cm on a side.
$$P\left( {1\,\,{\rm{hit}}\,\,{\rm{in}}\,\,{\rm{circular}}\,\,{\rm{region}}\,\,{\rm{in}}\,\,{\rm{one}}\,\,{\rm{throw}}} \right) = {{{S_{{\rm{circle}}}}} \over {{S_{{\rm{square}}}}}} = {{\pi \cdot {1^2}} \over {{3^2}}} = {\pi \over 9}$$
$$? = {\left( {{\pi \over 9}} \right)^2}\,\,\,{\rm{by}}\,\,{\rm{independency}}\,\,{\rm{on}}\,\,{\rm{throws}}$$

The correct answer is therefore (D).


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
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by Jay@ManhattanReview » Tue Oct 30, 2018 10:22 pm
BTGmoderatorLU wrote:Source: Economist GMAT

A game at the state fair has a circular target with a radius of 10 cm on a square board measuring 30 cm on a side. Players win prizes if they throw two darts and hit only the circular area on both throws. If Jim hit the board on both throws, what is the probability that Jim won the game?

A. 1/3
B. 1/9
C. π/9
D. π^2/81
E. π^2/9

The OA is D.
Area of the circle = πr^2 = π*10^2 = 100π cm^2
Area of the square = 30*30 = 900 cm^2

Probability of hitting ONE dart within the circle = (Area of the circle) / (Area of the square) = 100π / 900 = π/9

Probability of hitting both the darts within the circle = π/9* π/9 = π^2/81

The correct answer: D

Hope this helps!

-Jay
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by Scott@TargetTestPrep » Wed Oct 31, 2018 5:24 pm
BTGmoderatorLU wrote:Source: Economist GMAT

A game at the state fair has a circular target with a radius of 10 cm on a square board measuring 30 cm on a side. Players win prizes if they throw two darts and hit only the circular area on both throws. If Jim hit the board on both throws, what is the probability that Jim won the game?

A. 1/3
B. 1/9
C. π/9
D. π^2/81
E. π^2/9
T
he area of the circular target of the board is 100� cm^2 and the area of the entire board is 900 cm^2, so the probability of hitting the target once is 100�/900, or �/9. The probability of hitting the target twice is:

�/9 x �/9 = �^2/81

Answer: D

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