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

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A game at the state fair has a circular target with a radius  [#permalink]

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25 May 2014, 22:21
3
15
00:00

Difficulty:

85% (hard)

Question Stats:

57% (02:56) correct 43% (03:03) wrong based on 271 sessions

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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 at least one of the two attempts. What is the probability that Jim won the game?

a) 1-((9-π)/9)
b) 1-((18π-π^2)/81)
c) (9-π)/9
d) ((9-π)^2)81
e) (18π-π^2)/81
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Re: A game at the state fair has a circular target with a radius  [#permalink]

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26 May 2014, 03:23
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6
brunopanta wrote:
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 at least one of the two attempts. What is the probability that Jim won the game?

a) 1-((9-π)/9)
b) 1-((18π-π^2)/81)
c) (9-π)/9
d) ((9-π)^2)81
e) (18π-π^2)/81

The area of the circle is $$\pi{r^2}=100\pi$$;
The area of the square is $$side^2=900$$.

The probability of hitting the circle is therefore $$\frac{100\pi}{900}=\frac{\pi}{9}$$ and the probability of missing the circle is $$1-\frac{\pi}{9}=\frac{9-\pi}{9}$$.

The probability of hitting the circle on at least one of the two attempts = 1 - {the probability of missing on both of the two attempts} =

$$1-\frac{9-\pi}{9}*\frac{9-\pi}{9}=\frac{81-(9-\pi)^2}{81}=\frac{81-(81-18\pi+{\pi}^2)}{81}=\frac{18\pi-{\pi}^2}{81}$$.

Hope it's clear.
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A game at the state fair has a circular target with a radius of 10 cm  [#permalink]

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30 Jul 2015, 12:49
2
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 at least one of the two attempts. What is the probability that Jim won the game?

A. 1 - $$\frac{9-\pi}{9}$$
B. 1- $$\frac{18\pi-\pi^2}{81}$$
C. 9-$$\frac{\pi}{9}$$
D. $$\frac{(9-\pi)^2}{81}$$
E. $$\frac{18\pi-\pi^2}{81}$$
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A game at the state fair has a circular target with a radius of 10 cm  [#permalink]

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30 Jul 2015, 13:50
1
reto wrote:
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 at least one of the two attempts. What is the probability that Jim won the game?

A. 1 - $$\frac{9-\pi}{9}$$
B. 1- $$\frac{18\pi-\pi^2}{81}$$
C. 9-$$\frac{\pi}{9}$$
D. $$\frac{(9-\pi)^2}{81)}$$
E. $$\frac{18\pi-\pi^2}{81}$$

Fun question. Too bad I missed it on the first attempt.
To solve:
p(success) = 1 - p(failure), where p(failure) = 1 - (Area of Target/Area of Square)
p(success) = $$1 - (1-\frac{10^{2}\pi}{30^{2}})^{2}$$.
$$1-(1-\frac{\pi}{9})^{2}$$
$$1-(\frac{9-\pi}{9})^{2}$$
$$1-(\frac{81-18\pi+\pi^{2}}{81})$$
$$\frac{81-81+18\pi-\pi^{2}}{81}$$
$$\frac{18\pi-\pi^{2}}{81}$$

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Re: A game at the state fair has a circular target with a radius  [#permalink]

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31 Jul 2015, 07:40
Probability of Jim hitting the circular area at least once = 1 - (not hitting the circular area).

Area of the circular target = 100 (pi).

Area of the square board = 30*30 = 900 cm^2.

Probability of hitting the circle = 100 (pi)/900 = (pi)/9.

Probability of not hitting the circle = 1 - (pi)/9 = 9 - (pi)/9.

Probability => 1 - (9 - (pi)/9)^2 = 1 - (81 - 18(pi) + (pi)^2)/81 = 18(pi) - (pi)^2/81. Ans (E).
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Re: A game at the state fair has a circular target with a radius  [#permalink]

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28 Apr 2017, 07:33
brunopanta wrote:
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 at least one of the two attempts. What is the probability that Jim won the game?

a) 1-((9-π)/9)
b) 1-((18π-π^2)/81)
c) (9-π)/9
d) ((9-π)^2)81
e) (18π-π^2)/81

Does anyone else thinks that this is a poor question? - It is not stated that the player always hits at least square bord (30 cm side is pretty small). The player still can hit the wall on which the square board is hanging. Therefore to area with 1 probability is much bigger (and unknown) than 900cm2.
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Re: A game at the state fair has a circular target with a radius  [#permalink]

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16 Dec 2017, 04:20
I see generally the average time on such questions by Bunuel is near by2 mins, but i always find myself taking around 3 or more mins
is anyone else facing the same
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Re: A game at the state fair has a circular target with a radius  [#permalink]

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15 Mar 2019, 09:33
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Re: A game at the state fair has a circular target with a radius   [#permalink] 15 Mar 2019, 09:33
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