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SETS

by goyalsau » Tue Oct 05, 2010 6:38 pm
Set T consists of all points(x,y) such that x^2+y^2=1 . If point(a,b) is selected from set T at random, what is the probability that b>a+1?

1/4

1/3

2/3

1/2

3/5
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by limestone » Tue Oct 05, 2010 7:08 pm
Well, interesting problem, this is a circle's equation. But don't mind 'bout it.

First, x^2+y^2 = 1 then

x^2<= 1
y^2<= 1

Then

x^2<=|x|<=1 ( as x^2<=1, for example 0.2^2 <|0.2|<1 or 0.2^2 <|-0.2|<1 )
y^2<=|y|<=1

As x^2+y^2 = 1 and y^2<=|y| and x^2<=|x|

then |x| + |y| >= 1

From the above given information that b> a+1 or b - a >1 while x = a, b = y, or we can say y -x >1

For |x| + |y| >= 1 to become y -x >1 :
x must be negative so that |x| = - x
y must be positive so that |y| = y
then |x| + |y| >= 1 becomes -x + y>=1 or y - x >=1

For a pair of x and y; there're 4 cases of having sign:

x,y<0
x,y>0
x<0; y>0
x>0; y<0

Here only x<0; y>0 is acceptable. Thus P = 1/4

If you know about circle's equation, draw it out in coordinate plane, you'll get what I said.
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