PROJECT 4
Radioactive Decay Simulation
Background:
Radioactive decay is the process by which nuclei emit particles. When the nucleus of an atom emits
particles, the atom may no longer be the same element, but rather an isotope of some other element. It is
this process by which a radioactive substance decays into a non-radioactive substance.
In measuring how long it will take a radioactive substance to decay, the most important quantity is the halflife of the substance. The half-life is the length of time required (on average) for half of the radioactive
material present to decay. Depending on the element, a half-life might be a few microseconds or several
thousand years.
Consider the example of an element that has a half-life of one hour. Given an initial amount of this
material you can expect about half of it to remain after one hour. Observe that there is an astronomically
large number of atoms present. Over the course of one hour, approximately half of them decay. It is,
however, impossible to say which particular atoms will decay. The half-life being one hour means that for
a particular atom, there is a 0.5 probability that it will decay sometime within the first hour. This is true for
each individual atom.
This is the point of the simulation you are about to perform. Here, the original collection of m&ms
represents atoms. A shake represents one half-life. The m&ms that come up m are the ones that have
decayed.
Project:
Start with 30 or more m&ms (the more m&ms, the better). Place the m&ms in a cup. Shake thoroughly
and pour them out onto the table. Remove any m&ms that are showing m. Record the number of m&ms
left on the table. Pick up the remaining m&ms. Place them back in the cup, shake, and pour them out onto
the table. Again, remove those m&ms that are showing m. Record the number remaining. Keep repeating
this process until no m&ms are left.
Repeat the entire experiment five times. Keep records for each trial.
Terminology & Concepts:
Two events are called independent if the outcome of one event has not affect on the outcome of the other
event. In this case, whether a particular m&m comes up m or not m has nothing to do with the outcome for
any other m&m. In addition, the outcome of a particular m&m on one shake has no effect on the outcome
for that m&m on any subsequent shakes. For example, if a particular m&m comes up m on one shake, then
the probability that it comes up m on the next shake is still ½ (regardess of the outcome of the first flip).
The probability that two independent events both occur is the product of the probabilities that each one
occurs separately. Thus, the probability that a particular m&m comes up m twice in a row is
(1/2)(1/2)=1/4. Observe that this is the probability before any outcomes are known. Once the first outcome
is known, then the chance that the m&m comes up m on the second trial is again ½.
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Project 4
Radioactive Decay Simulation
Questions:
The following questions ask you to analyze the outcomes of your experiment. The goal is for you to
understand how basic probability theory affects the behavior of a large number of particles undergoing
radioactive decay.
In answering the questions, use complete sentences and proper punctuation. Your data should be displayed
in a neat organized tabular form. For each of the five experiments you should how the number of m&ms
remaining after each shake.
1.
How many m&ms remained after the first shake on each of your five simulations? Explain why you
did not always get half the amount you started with.
2.
What was the greatest number of shakes required to remove all the m&ms. Consider one of the m&ms
that survived all the way to the last shake. How many times in a row did that particular m&m come up
not m? What is the probability of flipping an m&m that many times and having it come up not m
every time? How do you explain the occurrence of such an event happening that has such a small
probability?
3.
How would the experiment be different if you doubled the number of m&ms. How many more shakes
on average would you expect to be necessary in order to remove all the m&ms?
4.
Since the half-life of a single m&m is one shake, how many shakes would it take (on average) for a
mole of m&m to completely decay. (Note: A mole of m&ms would consist of approximately
6.02 10 23 m&ms.) Estimate the answer by assuming that after each shake, half the pennies are
removed.
5.
Determine the average number of m&ms remaining after each shake by combining the results of your
five simulations. Use theses values to make a data table that shows the number of m&ms remaining as
a function of the number of shakes. Carefully plot theses data points and smoothly connect them.
What type of functions does your graph resemble: polynomial, exponential or logarithmic?
6.
Find a formula that approximately gives the graph you sketched. Use one of the following general
forms: y  ax x , y  AB x , y  a log b x where y represents the number of m&ms remaining after x
shakes.
7.
Some radioactive elements do not decay to a stable element in one step. Rather, they decay into
another radioactive element that then decays into another element (which still may be radioactive). A
well-known example of this is the radioactive element uranium which undergoes several decay
processes before becoming lead (a process that takes billions of years even for a small amount of
uranium.
Suppose that we wish to investigate an element that must undergo two decay processes in order to
reach a stable element. Using the m&ms to model atoms, how could you modify the experiment to
simulate this two-step decay process? Describe your experiment and then implement it several times
using eight m&ms (atoms). Compute the average number of flips required to remove all the m&ms.
How does this number compare to the average number of flips required to remove eight m&ms in a
single step radioactive decay? Make a conjecture about the average number of flips required to
remove all the m&ms in modeling a k-step radioactive decay process.
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