In the coordinate plane, rectangular region R has vertices a

Question

In the coordinate plane, rectangular region R has vertices at (0,0), (0,3), (4,3), and (4,0). If a point in region R is randomly selected, what is the probability that the point's y-coordinate will be greater than its x-coordinate?

7/12
5/12
3/8
1/3
1/4

A. 7/12
B. 5/12
C. 3/8
D. 1/3
E. 1/4

Bunuel · Accepted Answer

See the diagram below. https://gmatclub.com/forum/download/file.php?id=13394 Now, rectangle R has an area of 3*4=12. All point that has y-coordinate greater than x-coordinate lie above the line y=x , so in yellow triangle, which has an area of 1/2*3*3=4.5. So, the probability equals to favorable outcomes/total=yellow triangle/rectangle R=4.5/12=3/8 . Answer: C. graph.PNG

shrive555 · Answer

i took that y intercept 3 to be greater than x all coordinates and divided by 12 and got 1/4. which is completely wrong. and according to your solution in that yellow region every point is greater than x not just only y intercept. Great.

Bunuel · Answer

Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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Narenn · Answer

I think Bunuel's above explanation is most illuminating and detailed; I do not find any alternate way for the explanation. +1 for that great solution.

Paris75 · Answer

Hi,

this is how I did it:

The probability if X=3 that Y>X is 0
The probability if X=2 that Y>X is 1/4
The probability if X=1 that Y>X is 2/4
The probability if X=0 that Y>X is 3/4

Then:  1/4*1/4 + 1/4*2/4 + 1/4*3/4 = 6/16 = 3/8

Hope it helps

bharatdivvela · Answer

there is no better solution than this

compute the area above y=x line and below y=x line and then proceed to finding probability

AKG1593 · Answer

I calculated that the no. of points in the region including the boundaries of the rectangle= 16 .
Out of that  6  points  6  ordered pairs- (0,1),(0,2),(0,3),(1,2),(1,3)  and  (2,3) ],will have  x  coordinate less than  y  coordinate.
So probability= 6/16=3/8 
Isn't this approach right?

Bunuel · Answer

No, this approach is not right. A point has no dimension, hence there are infinitely many points in any area/segment. The problem with your solution is that you assume that the coordinates must be integers, which is nowhere given.

Hope it's clear.

m3equals333 · Answer

Hi Bunuel...I came across this question and kept getting caught up by the y=x line.

If we calculate the isoc triangle's area, wouldn't this include all points on the y=x line (on which y is not > x)?

Bunuel · Answer

A line has no area, hence there won't be any difference.

m3equals333 · Answer

ah ha, that clears it up, thanks a bunch

Bobbyvg · Answer

Hi Brunei,

How the rectangle has 16 points (if we calculate the borders ). Isn't it should be 20 . e.g (0,0),(1,0),(2,0),(3,0),(4,0).
5 points in 1 row and 4 rows all together 5*4 = 20 . Am I missing something ?

Bunuel · Answer

I don't understand why are you concerned about the number of points? The solution talks about the area...

Bobbyvg · Answer

Brunei ,

I didnot get your point(in earlier quote) why calculating points is wrong (I know we get an incorrect answer ). Isn't Probability = No of successes(condition)/ Total no of outcomes and if we go by points arenot we just following it .

Bunuel · Answer

(Probability) = (Favorable outcomes)/(Total) = (The area of yellow triangle)/(The area of the rectangle). Notice that we have an area there no the number of poiint whose coordinates are integers only.

Check other  Probability and Geometry  questions in our  Special Questions Directory .

Hope it helps.

Mbawarrior01 · Answer

I did graph the region correctly. However, when I looked at the region I thought the selected region is slightly less than half.

However, the answer choices are quite close. So you have to calculate the area, no choice.

austindembeck · Answer

I do not understand or cannot recalculate the 1/2*3*3=4.5. How did you get 1/2 as other side?

Fdambro294 · Answer

the big Trap to avoid is to remember that the Coordinates do NOT have to Be Integers.

After plotting the Points, you have a Rectangle with Length = 4 Units and Width = 3 Units

the Line given by the Equation: y = x ---- will have the X-Coord. EQUAL to the Y-Coord. on every Space on that Line

Every Coordinate Point ABOVE the Line y = x will have a Y-Coord. that is GREATER than the X-Coord.

So basically we are looking for the following:

(Area of the Region that is Y > X but INSIDE the Rectangle) / (Entire Rectangular Region) = Probability that a given Coordinate will have its Y-Coordinate Greater than its X-Coordinate

the Line y = x will come through Vertex (0 , 0) and pass through Point (3, 3) on the TOP Length of the Rectangle.

the Area above this line but Inside Region R of the Rectangle will be Defined by the Right Triangle with the following Vertices: (0 , 0) ; (0 , 3) ; (3, 3)

Area of the Right Triangle = 1/2 * 3 * 3 = 9/2

Area of the Entire Rectangular Region R = 4 * 3 = 12

Probability = (9/2) / (12) = 9/24 = 3/8

-C-

Crytiocanalyst · Answer

Area of the rectangle = 12

Area of the region where y>x = 1/2*3*3

Therefore the required probability = 9/24 = 3/8

Thererfore IMO C

In the coordinate plane, rectangular region R has vertices a

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In the coordinate plane, rectangular region R has vertices a

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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My Rewards