Another way to look at it:
0.0<=x<0.5->Math.round(x)=0
0.5<=x<1.5->Math.round(x)=1
1.5<=x<2.5->Math.round(x)=2
2.5<=x<3.5->Math.round(x)=3
3.5<=x<4.5->Math.round(x)=4
….
MAX-0.5<=x<MAX->Math.round(x)=MAX

From this we see that the range for 0 and MAX is only 0.5, while the between numbers have a range of 1, and thus it is twice as likely to get a middle number than one of the ends.

I’m sure you’re not daft. It may be that I confused matters by only going to up three in my diagrams, when my ActionScript code shows a range that goes higher. Six parts was just arbitrary, otherwise the image would have been wider.

The first diagram is intended to show that the two end numbers — no matter how long the sequence — are only chosen half as often as the middle numbers. In my ActionScript, the return value of Math.random() * 9 might actually be, say, 9.7. When rounded with Math.round(), that would become 10. The return value might be 8.5, which would become 9. Might be 9.3, which would become 9. The number 9 has an unfair advantage over 10, in this case.

Make sense? If not, let me know.

I really appreciate your explaining this topic. It’s something I’ve always shied away from because I couldn’t see the use.