Perfectly flat probability
The continuous uniform distribution is the "no favourites" model: a value is equally likely to land anywhere between a and b, a bus arriving at a random moment in its interval, a spinner's stopping point. Its density is a rectangle of height 1/(b−a), and probability is literally area: the share of the width your interval covers.
X uniform on [2, 10]. P(3 < X < 7) = 4 ÷ 8 = 0.5: the middle half of the width carries half the probability. Mean 6, SD 2.31. Nothing concentrates anywhere: the interval [2, 3] and [9, 10] are equally probable at 0.125 each, an property no bell curve will ever give you.
Asking for P(X = exactly 4.5). On any continuous distribution a single point has width zero and probability zero: only intervals carry probability. The question is always about ranges.
Frequently asked questions
Why is the SD (b−a)/√12?
It falls out of integrating (x − mean)² against the flat density. The practical takeaway: a uniform's SD is about 29% of its full width, much fatter, relatively, than a normal's.