It is really good that you had a go AND that you stated your lack of confidence.
It is not a really easy question... I'll give it a go
First, like you said, the big cube is cut into 5*5*5
Starting with the top row and working down, I am going to count how many red sides each little cube has.
Top and bottom rows
3*4+2*12+9 = 12+108+9 = 129 red sides on top row and 129 on the bottom one too
3 rows in the middle
3*(2*4+1*12) = 60 other red sides
Total number of red sides = 129+129+60 = 318 red sides altogether
There are 125 little cubes so these have a total of 125*6 = 750 sides
So if you choose a little cube at random and roll it I think the probability of rolling a red is 318/750 = 53/125
I think this is correct but there might have been an easier way to do it. I am really not sure. :)
CPhill has assumed that this is a paralellogram, I assumed that it did not have to be.
Mine failed because i did not have all the information?
Was it a parallelogram?
Perhaps the question (not you) forgot to mention this?
If it did not have to be a parallelogram, or if you were suposed to prove that it had to be a parallelogram, then CPhill's answer is not valid, or at least not complete. I suppose CPhill has stated his assumption in the first line.
I also acknowledge that we do a lot of maths detective work in this forum so I shall congratulate CPhill for assuming a (possible) fact that was not given.
I guess even in my attempt I did assume that P, E and S were collinear points, I guess there is no great reason why I should have assumed this either.
I am glad that CPhill helped you Juriemagic :))
We always welcome bright, thinking students like yourself here :)
Oh, I know you are really busy and there is such a lot for you (and every other student) to learn.
But GeoGebra is a great program and a lot of fun to use. The diagram I made was interactive so I could move the points around and still meet all the given criterion. It's a lot of fun to use a program like that, and it helps a lot when trying to see what is happening with these types of problems.
I tried drawing it and I found that only very specific angles will work (in order to make PES a straight angle.)
I randomly chose an x angle and then constructed it from there (using GeoGebra)
This is what I got - only in GeoGebra you can move things to try and make it better.
As you can see, even with the 72 degrees, PES is not a straight angle.
So obviously only a limited number of x angles can work (maybe only one)
I have not gotten further than this yet but the focus of my thoughts is what angles are necessary in order to make PES a straight angle.......