Law of Radioactive Decay 

  The rate of decay of a radioactive substance is directly proportional to the amount of the substance you have, or 

Integrating the equation above and get the equation for the amount of a substance as a function of time:

Solving for the time it takes for 1/2 how the substance to decay is: 

 

Radioactivity 

The amount of decays that happens every second is refereed to as activity  

For initial activity of a substance the equation: 

Applying initial activity to all of the other equation we get the following equations: 

Data Table provided by The science teacher program.

Half life 

Half life is a mathematical term, a statical term.  Because the random nature of decaying isotopes we use statistical models to gain understanding of the phenomenon.  The half life of a radioactive material is approximate amount of time it takes for a sample of material to be transmuted.  And like all statistic models (at least good ones) the larger the sample size, the closer to the actrual number.   

Half life and Activity 

Diagram provided by HyperPhysics

Think of it this way if you had two tons of plutonium 239, and you had all the necessary equipment to "count" how much plutonium you had after 2.44 x 10^4 years you would have one ton left (+ or - a couple of grams, maybe a kilogram).   Now do the same experiment but use only two atoms, there are no guarantees that after one single day or even a single second that there would be two plutonium atoms.  On the flip side 60,000 are even 80,000 years could pass and none of the plutonium atoms have decayed.   It's a random process that we apply a mathematical model too, for large amounts it works great, the smaller and smaller the amount the less accurate it becomes, remember that.