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10 Mar 2008, 09:45
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Line L passes through (5,0) and (0,10).If a point is selelcted at random from the triangular region encircled by axis-x, axis-y, and line L, what is the probability that y<x?
Please kindly provide explanation.
Please kindly provide explanation.
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10 Mar 2008, 10:01
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I am not sure about exact answer.
Area of Whole traingle = 1/2*5*10 = 25
Area of triangle Between Y > 5 and Y <= 10 = 1/2*5*5 = 12.5
In second traingle all the values of Y will be greater than X so Prob. for Y > X = 12.5/25 = 1/2
However there will be case in which Y>X even within the lower triangle so I am not sure how to handle them.
E.g. (1,4) is valid point in lower triangle here Y>X so such probability also needs to be accounted for.
I can say that probability of X>Y will be around 1/4. So I will go by the answers rather than doing the whole calculations.
Area of Whole traingle = 1/2*5*10 = 25
Area of triangle Between Y > 5 and Y <= 10 = 1/2*5*5 = 12.5
In second traingle all the values of Y will be greater than X so Prob. for Y > X = 12.5/25 = 1/2
However there will be case in which Y>X even within the lower triangle so I am not sure how to handle them.
E.g. (1,4) is valid point in lower triangle here Y>X so such probability also needs to be accounted for.
I can say that probability of X>Y will be around 1/4. So I will go by the answers rather than doing the whole calculations.
10 Mar 2008, 16:58
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As Abhijit said, the whole triangle is 25. The fraction of this larger triangle having points where y<x are contained in the smaller triangle whose sides are the x axis, the line having points (0,10) and (5,0), and the line y=x. The line y=x intersects the line with points (0,10)(5,0) at (10/3, 10/3):
y-10=(-10/5)(x-0) (Point slope form of line between 0,10 and 5,0)
y-10=(-2)(y) (sub y=x)
3y=10
The area of the smaller triangle is 1/2 * 5 * 10/3 = 25/3
The ratio of the smaller triangle to the larger is (25/3)/25 = 1/3
The probability is 1/3 that y<x
y-10=(-10/5)(x-0) (Point slope form of line between 0,10 and 5,0)
y-10=(-2)(y) (sub y=x)
3y=10
The area of the smaller triangle is 1/2 * 5 * 10/3 = 25/3
The ratio of the smaller triangle to the larger is (25/3)/25 = 1/3
The probability is 1/3 that y<x
