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Combining Probabilities: Instructor's Guide

Main Ideas

  • Statistical mechanics
  • Indexing
  • Probabilities of uncorrelated systems

Students' Task

Estimated Time: 20 minutes

  1. Students are broken into small groups and presented the following scenario: “Consider two systems A and B, which you can think of as dice or quantum mechanical systems with several eigenstates. Each system has a set of possible states i. If the systems are uncorrelated, we can represent their separate probabilities as $P_{i}^{A}$ and $P_{j}^{B}$.”
  2. The groups must answer together several questions about the probabilities concerning the scenario.

Prerequisite Knowledge

  • Familiarity with summation notation and indexing of terms
  • Basic understanding of probabilities and statistics is preferable


Activity: Introduction

This activity works well as a gauge for how familiar students are with probabilities in statistical mechanics and needs little introduction. A drawing of the scenario can be placed on the board for visual aide; one possible diagram would be two boxes, one box A and another box B, that each have i number of possible eigenstates with probabilities $P_{i}^{A}$ and $P_{j}^{B}$. Noting that an analog of this scenario would be the rolling of two dice with i sides will provide students a more physically graspable example. After giving the scenario to the students, the instructor can then ask:

What is the probability of system A being in state i and B being in state j?

After discussion and presenting the answer, the instructor can then ask:

What is the probability of system A being in state n regardless of what the state of B is?

Activity: Student Conversations

Many answers students give can be addressed by asking questions to which they know the answers. For instance, when students answer that the probability is the sum of the two probabilities, you can ask what the probability when rolling two six-sided dice is of getting a seven and a one. A related question (also useful) is to ask if they have two systems, each of which has $N$ states, how many states does the combined system have?

Activity: Wrap-up

For this activity, students will typically know what the answers are, but will have a difficult time presenting the answers in a mathematical language. After the class has determined that the probability of finding box A in state i and box B in state j, write on the board that this probability is notated by

$$ P_{ij}=P_{i}^{A}P_{j}^{B} \; \; \; .$$

Now, if this is the probability of finding box A in state i when box B is in state j, then the probability of finding box A in state n when box B is in any state will be a sum of the probabilities of finding box A in state n for each of box B's possible states. Write on the board that this is expressed mathematically as

$$P_{n}^{A}= \sum_{j}P_{nj} \; \; \; . $$

If this problem is presented as a situation with two six-sided dice, some students may explicitly write the addition of each of the probabilities for their answer. Note to the class that as they progress through the statistical mechanics section of this course, they will find that leaving terms in summation notation until all simplifications have been performed will make problems much easier to solve. Even if this notation seems unnecessary or hassling, assure the class that it will pay off when problems become more complex.


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