Just wondering, if decay is random, why does the activity half every half life, as in, why does it have to reduce by the same proportion in the same time period?

I was studying about light waves today and and in the introduction of the chapter there is a paragraph as follows:- 1) The Dutch physicist Christian Huygens suggested that light may be a wave phenomenon. 2) The apparent rectilinear propagation of light may be due to the fact that the wav...

I've noticed in my physics textbook (and in a lot of other popular sources), that the process of integrating non-constant acceleration to get to a velocity formula, the integrating bounds imposed on the velocity-part seem wrong. In the above snippet, they're using $v_0$ and $v$ as bounds in th...