Chem 146
Lab 3:
I.
Spring 2002
Statistical Distributions
Introduction
Many texts on nuclear theory never explain how the distributions that are supposed to
describe nuclear decay are connected to the nuclear decay process. In this lab we will
explore the three most commonly used statistical distributions: Binomial, Poisson, and
Gaussian.
II.
Reading Material
Further information can be found in Knoll, Chapter 3.
III.
Binomial Distribution
As an example of how to apply these distributions to nuclear decay, let's look at the
binomial distribution. This distribution is applied when there are two possible outcomes
for something, like a coin comes up either heads or tails when flipped or a nucleus either
decays or it doesn't. It is most useful when there are a small number of trials. The form
of the binomial distribution is:
n!
Pb (m) 
p m (1  p) n m
m!(n  m)!
where P(m) is the probability of having m successes in n tries, n is the number of trials,
m is the number of successes, and p is the probability of succeeding.
The mean of the binomial distribution is: n  np
The square of the standard deviation is:  2  np(1  p)
For the situation of flipping a coin, we'll consider the coin coming up heads as a success.
Then, n is the number of times the coin is flipped. m is the number of times the coin
lands up heads, and p is 0.50, since the chance of getting a head is equal to the chance of
getting a tail. If we flip a coin ten times, the probability that a head comes up 7 times is
10!
(0.50) 7 (1  0.50)107  0.117
7!(10  7)!
In the case of nuclear decay, we consider it a success if the nucleus undergoes decay. In
the expression for the binomial distribution, n is the total number of nuclei in a sample.
m is the number of nuclei that decay. p is the probability of undergoing decay in a
specified length of time.
If it is known that for a collection of 1000 nuclei, 600 on average will decay in one
second, then the probability of decay in one second is 0.6. If I have a sample of 10 of
those nuclei, the mean number that will decay in 1 second is np  10  0.6  6 . The
standard deviation is
  n (1  p)2  6(1  0.6)2  1.5
1
1
As an example, let's calculate some probabilities. What is the probability of only one of
the ten atoms will decay in one second?
10!
Pb 1 
(0.6)1 (1  0.6)101  0.0016
1!(10  1)!
What about the chance of exactly four of the ten atoms will decay during one second?
10!
Pb 4 
(0.6) 4 (1  0.6)104  0.11
4!(10  4)!
What are the chances that exactly five of the ten atoms will decay in one second?
10!
Pb 5 
(0.6) 5 (1  0.6)105  0.20
5!(10  5)!
What is the probability of six of the atoms will decay? Remember that 6 is the average.
10!
Pb 6 
(0.6) 6 (1  0.6)106  0.25
6!(10  6)!
We see from this example that even though the most probable number of decay events in
one second is 6, the probability of this happening is only 25%. The binomial distribution
is very useful for a small number of radioactive nuclei, but you'll notice that as n gets
big, this binomial distribution becomes a real mess.
IV.
Poisson Distribution
As the number of atoms being considered gets large and the time being considered
remains small compared with the half-life of the species, we can approximate the
binomial distribution with the Poisson distribution:
(np) m
(n ) m
Pp (m) 
exp( np) 
exp( n )
m!
m!
The mean square of the Poisson distribution is n  np .
The square of the standard deviation is  2  np  n .
V.
Gaussian Distribution
A further approximation of the binomial distribution is the Gaussian distribution.
1
1 (np  m) 2
1
1 (n  m) 2
Pg (m) 
exp( 
)
exp( 
)
2
np
2 2
2np
2 
The mean of the Gaussian distribution is n  np .
The square of the standard deviation is  2  np  n .
These are the same values in the Poisson distribution.
VI.
Prelab
In your notebook, write down the relevant decay information for the isotopes we may use
in this lab. They are identified in section VII.B.1 (below). Include decay mode(s), half
life, prominent γ lines, and calculate the Q-value(s).
VII.
Experimental
This experiment will be divided into two parts. The first section will give you experience
with the Binomial distribution. In order to keep the number of trials to a number small
enough to allow for manual calculations, you will perform experiments by flipping coins
and counting the number of heads that come up during each attempt. In the second part
of the experiment, you will be taking data from a radioactive source. Theoretical decay
distributions will be calculated from the Gaussian distribution for these data.
A. Coin Tossing
1. Each group will be given a collection of 17 pennies.
2. Flip all of the pennies and record how many of the coins land "heads up".
Perform this 100 times.
B. Radioactive Decay
1. Set up the counting system with either the 60Co, 133Ba or 137Cs source, in such
a manner that you will record between 50 and 150 counts with the scaler in a
counting interval of 1 second. You will have to play around with the distance
between the source and the detector and with the discriminator to get this
right.
2. Record the number of counts for at least 100 observations. Remember, you
get closer to the true value as you perform more experiments.
3. Be sure to determine the background count rate to be subtracted from your
recorded values. Count the background using the same discriminator settings
as when you counted your sample at least 100 times.
VIII. Report
A. Coin Tossing
1. Plot your data from the coin flipping experiment in three different histograms.
All three graphs will be histograms with "number of heads in a trial" along the
x-direction and "percent of trials" in the y-direction. "Percent of trials" is the
number of trials that contained a particular number of heads divided by the
total number of trials. The difference between the three will be the bin size
used along the x-axis.
2. a.) Use a bin size of 1. b.) Use a bin size of 2. c.) Use a bin size of 3.
3. How does the different bin size affect the display of the data?
4. Plot an ideal binomial distribution over the histogram of bin size of 1. How
well does the theoretical distribution fit the experimental data?
5. Determine the theoretical mean and the experimental mean. Calculate the
theoretical standard deviation. What fraction of the observations are outside
one standard deviation?
B. Radioactive Decay
1. Plot your data in two histograms this time in the same manner as above.
2. a.) Use a bin size of 1. b.) Use a bin size of 10.
3. How does the different bin size affect the display of the data?
4. Find the experimental mean for your data set. Superimposed on the plot with
bin size of ten, plot the appropriate Gaussian distribution. How well does the
theoretical curve fit the data? What fraction of observations are outside one
standard deviation? What do you think would happen to the fit if you
recorded more measurements?