Statistics for loaded dice
Here is the solution to these problems
Figure 1: Face loaded die that will be used in this project.
Particles with kinetic energy \( E_k \) move in a "landscape" of states \( i \) with potential energies \( E_i \). The probability of the particle to be in state \( i \) is
$$ \begin{eqnarray} P_i&=&\frac{1}{Z}e^{-\beta E_i}\\ Z&=&\sum_{i=1}^Ne^{-\beta E_i}, \end{eqnarray} $$where \( \beta=\frac{\alpha}{E_k} \), \( \alpha \) is a positive proportionality constant and \( N \) is the number of states.
The normalization factor \( Z \) ensures that the sum of the probabilities is \( \sum_iP_i=1 \).
The relative probability of two states is
$$ \begin{equation} \frac{P_i}{P_j}=e^{-\beta(E_i-E_j)}. \label{_auto1} \end{equation} $$
When we throw a die that rolls many times along a flat surface it has a typical kinetic energy \( E_k \) just before settling with a given face up. When the die is lying still with a face \( i\in{1,6} \) up it is in a potential well with energy \( E_i \). A homogeneous die has the probability \( P_i=\frac{1}{6} \) and the same potential well depth \( E_i \) for all the 6 faces \( i\in{1,6} \). A face loaded die has one face that is heavier than the others. This will affect the probabilities \( P_i \).
Do face loaded dice follow the Boltzmann conjecture?
Hint. We do not know which face \( i\in\{1\ldots 6\} \) is loaded. In order to have a naming convention that is the same for all cases we will define the loading in the following way:
As an example we can say that face 2 is loaded and then face 5 is opposite (the sum of opposite faces in a die is always 7). Let us give the positions of the die names according to which faces turns up:
- \( d \): loaded face down, thus \( d=5 \)
- \( s \): loaded face to the side, thus \( s=\{1,3,4,6\} \)
- \( u \): loaded face up, thus \( u=2 \)
We will name the energy differences as follows:
$$ \begin{eqnarray} \Delta E_{ud}&=&E_u-E_d\\ \Delta E_{us}&=&E_u-E_s\\ \Delta E_{sd}&=&E_s-E_d \end{eqnarray} $$
a) Find expressions for the potential energy differences of the different faces of a loaded die.
b) What does the corollary to Boltzmanns conjecture predict for the relative probabilities of the different faces pointing up?
c) Find expressions for the normalization factor, \( Z \) of the die and for each probability.
d) Plan and execute an experiment with a loaded die to test if the Boltzmann conjecture is valid for face loaded dice.
Hint. Note the result every time you throw the die. In this way you conserve the maximum amount of data from your experiment and
- you can reuse the data if you get new ideas for how to analyse it
- you can share and pool your data with that of others to get better statistics
e) Do face loaded dice follow the Boltzmann conjecture?
f) Can the use of the Boltzmann conjecture improve estimation of the probabilities of landing on each side of a loaded die?