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Numbers x and y are chosen independently and uniformly at random from

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Math Expert
Joined: 02 Sep 2009
Posts: 59590
Numbers x and y are chosen independently and uniformly at random from  [#permalink]

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13 May 2019, 23:30
00:00

Difficulty:

95% (hard)

Question Stats:

31% (01:46) correct 69% (02:18) wrong based on 55 sessions

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Numbers x and y are chosen independently and uniformly at random from the interval [0, 1]. Which of the following numbers is closest to the probability that x, y, and 1 are the side lengths of an obtuse triangle?

(A) 0.21
(B) 0.25
(C) 0.29
(D) 0.50
(E) 0.79

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Joined: 19 Oct 2018
Posts: 1158
Location: India
Re: Numbers x and y are chosen independently and uniformly at random from  [#permalink]

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24 May 2019, 18:33
2
We need the obtuse triangle whose sides are x,y and 1
As x and y lies in the interval of [0,1], the longest side of triangle must be 1
x^2 + y^2 < 1 {inequality holds for obtuse triangles)
Also we know that sum of two sides of triangle must be greater than the third side
x+y>1

We can use geometric probability to get our desired results.

Desired area is the area lie between two curves x^2 + y^2 < 1(circle of radius 1) and x+y>1(straight line)
Total area= square of side length 1

As we can see in the figure, Desired area(red portion)= {1/4(pi*1^2)}-(1/2*1*1)=pi/4-1/2
Total area= 1*1=1
The probability that x, y, and 1 are the side lengths of an obtuse triangle= (pi/4-1/2)/1=(pi/4-1/2)

Bunuel wrote:
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Intern
Joined: 11 May 2014
Posts: 3
Re: Numbers x and y are chosen independently and uniformly at random from  [#permalink]

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30 May 2019, 03:37
Can someone explain me the solution of this question. Not able to get the anser.
Intern
Joined: 16 Feb 2019
Posts: 6
Re: Numbers x and y are chosen independently and uniformly at random from  [#permalink]

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30 May 2019, 12:10
What is geometric probability ? Never heard of this term before .

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VP
Joined: 19 Oct 2018
Posts: 1158
Location: India
Re: Numbers x and y are chosen independently and uniformly at random from  [#permalink]

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30 May 2019, 12:23
Okay i'll give you an example. There is dart circular board of radius 4cm. If you hit inside a concentric circle of radius 0.5 cm, that lies inside the dart board, you will score 25. What is the probability that you will score 25 on the first shot, if you hit the dart board.

Probability=Area of smaller circle/Area of bigger circle= (pi/4)/(16pi)=1/64

geometric probability is used when you can measure the outcomes in length, area or volume.

Ayushkumar2294 wrote:
What is geometric probability ? Never heard of this term before .

Posted from my mobile device
Re: Numbers x and y are chosen independently and uniformly at random from   [#permalink] 30 May 2019, 12:23
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