The chance that any penny will come up tails on any toss is always the same, 50 percent. However, once a penny has come up tails, it is removed. Thus, about half the pennies are left after the first toss.
Even though half of the remaining pennies come up tails on the second toss, there are fewer pennies to start with. After the first toss, about 1/2 of the original pennies are left; after the second, about 1/4; then 1/8, 1/16, and so on. These numbers can be written in terms of powers, or exponents, of 1/2: (1/2)1, (1/2)2, (1/2)3, and (1/2)4. This type of pattern—in which a quantity repeatedly decreases by a fixed fraction (in this case, 1/2)—is known as exponential decay (click to enlarge photo below).
Each time you toss the remaining pennies, about half of them are removed. The time it takes for half of the remaining pennies to be removed is called the half-life. The half-life of the pennies in this model is about one toss.
If you’re using painted wooden cubes, the probability that a cube will land red side up is 1/6. (Each cube has six sides, and only one of those sides is painted red.) It takes three tosses for about half the cubes to be removed, so the half-life of the cubes is about three tosses. After one toss, 5/6 remain; after two tosses, 5/6 of 5/6, or 25/36, remain; and after three tosses, (5/6)3 = 125/216 of the cubes are left.
Tossing the coins or cubes is an unpredictable, random process. Rarely will exactly 1/2 of the coins or 1/6 of the cubes decay on the first toss. However, if you repeat the first toss many, many times, the average number of coins that decay will approach 1/2 (or cubes that decay will approach 1/6).
In this model, the removal of a penny or a cube corresponds to the decay of a radioactive nucleus. The chance that a particular radioactive nucleus in a sample of identical nuclei will decay in each second is the same for each second that passes, just as the chance that a penny would come up tails was the same for each toss (1/2) or the chance that a cube would come up red was the same for each toss (1/6).
The smaller the chance of decay, the longer the half-life (time for half of the sample to decay) of the particular radioactive isotope. The cubes, for instance, have a longer half-life than the pennies. For uranium 238, the chance of decay is small: Its half-life is 4.5 billion years. For radon 217, the chance of decay is large: Its half-life is one thousandth of a second.