Entropy Paradise
Work with a partner for this activity. You will share one pair of dice but will each need your
own clicker.
1. Each partner: Select a number between 1 and 6. Record your number ______. Each person
simultaneously rolls a die. Keep track of the number of times you have to roll your die to obtain
your number. The first time your number appears, you win!
a) Does either partner have a better probability of winning or losing, based on the number they
selected? _____
b) What is the probability of a given number appearing on any roll of the die? _____
c) If each of you roll one die 10 times, and add the numbers, in theory, what is the probability
you and your partner will have the same total score? ______
d) Each partner: Roll one die 10 times and add the numbers. What is your total score?
Partner A ______
Partner B ______
Compare with your neighbors: Neighbor A ______ Neighbor B ______
e) If you roll the die 100 times and add the numbers, in theory, what is the probability you and
your partner will have the same score? ______
That was a boring game. Let’s make it more interesting by using two dice, one red and one
white.
2. List the possible numerical outcomes when two dice are rolled and the numbers summed.
3. Each partner: Roll the pair of dice.
a) What was the sum of the two dice for your roll? ________
b) When prompted, submit your value to the class data set using your clicker.
c) Sketch the graphs of the distributions of outcomes.
d) Out of the approximately 300 dice rolls today, what is the most probable and least probable
outcome (sum of the two numbers)? __________
__________
most probable
least probable
e) Why? Explain by writing out the different ways the most probable value could be obtained
by rolling a pair of dice.
Red:
White:
f) What is the total number of configurations that will give the most probable score? _______
Entropy Paradise
g) Next, compare this to the number of different ways the values of 3 and 9 could be obtained.
Configurations to get a sum of 3:
Red:
White:
Configurations to get a sum of 9:
Red:
White:
h) Which total value has the most configurations: 3, 9 or the most probable value?
4. Each particular combination of numbers on the dice is called a microstate. How many
microstates exist for a pair of red and white dice? _______
The sum of the numbers on the dice is called a macrostate. Consider the class results from the
two-dice exercise.
What is the most probable macrostate? _______
What is the correlation between the number of microstates and the probability of a given
macrostate? Complete this sentence:
As the number of microstates increases, the probability of a given macrostate _____.
5. The probability of a given macrostate is given by the number of microstates that will produce
the macrostate (called the multiplicity, or Ω) divided by the total number of possible
microstates. Complete the table below to see how this works.
Total probability =
Entropy Paradise
The 2nd Law of Thermodynamics can be stated as follows: The equilibrium state of an isolated
system is the one in which the number of possible microscopic states in the largest.
It can be restated as: An isolated system tends toward an equilibrium macrostate with maximum
entropy, because then the number of microstates is the largest and this state is statistically most
probable.