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Exponential decay of a radioactive substance

One of the most important characteristics of radioactivity is that it decays exponentially. 
This has two basic mathematical implications at this level. 
1 The rate falls by a constant ratio in a given time interval. The time it takes to fall by a half is always the same. It also falls to a tenth in equally regular, but longer, time intervals. 
2 The rate of decay is proportional to the amount that is left. This can be seen in the experiment to model radioactive decay. The number of coins that decay in any ‘shake’ is proportional to the number that is left. 
From these features, you can argue, respectively, the following points. 
The chance of an atom disintegrating is constant in time. Radioactive decay is a series of many chance events, all with an unalterable chance. 
The rate of disintegrations is proportional to the total number of unchanged radioactive atoms at that moment. Both the rate and the stockpile itself die away exponentially with the same characteristic half-life.