Two dimensional sample space

[mdnazmulh]

Let’s select two numbers x and y such that 1<=x<=4 and 2<=y<=6. What is the probability of x +y>=5?
I solved the problem and attached it. Please check my solution and tell me whether my approach of solution is correct or wrong. The most puzzling matter is that I'm getting two different answers. Any suggestion would be of great help to me regarding my attempted solution. Thanks.

 

[mathman]

The thumbnails are almost unreadable.

 

[diggy]

If I'm reading what you are doing correctly, it looks like the sample space you drew is an "or" rather than an "and". That is your first picture looks like 1<x<4 "or" 2<y<6. If you do an "and" instead then you should get a box, not a tetris piece.

 

[Bacle]

I think if you graph all three equations, you will see a geometric way
of getting the answer, and use integration to do it.

 

[blue_raver22]

Your approach to solving the problem is correct. The probability of x+y>=5 can be calculated by finding the area of the region where x+y>=5 and dividing it by the total area of the sample space.

In this case, the total area of the sample space is (4-1)(6-2) = 12 units^2. The region where x+y>=5 lies above the line y=5-x in the sample space, as shown in the attached solution.

To find the area of this region, we can use the formula for the area of a triangle, which is (1/2)(base)(height). In this case, the base is (4-1) = 3 units and the height is (6-5) = 1 unit. Therefore, the area of the region is (1/2)(3)(1) = 1.5 units^2.

The probability of x+y>=5 is then calculated by dividing the area of the region by the total area of the sample space, which gives us 1.5/12 = 0.125 or 12.5%.

It is possible that you are getting two different answers due to rounding errors or a mistake in your calculations. I would suggest double-checking your calculations and making sure you are using the correct formula for finding the area of the region.