# Statistics: Degeneracy and Multiplicity

• teme92
In summary, when considering drawing one card from a deck with no jokers or other special cards, the number of microstates is equal to the number of card types (Ace, 2, 3, ..., King) which is 13. The number of macrostates is equal to the number of suits, which is 4. The degeneracy of the macrostate "spade" is equal to the number of cards in the spade suit, which is 13. The multiplicity of the macrostate "King" is equal to the number of Kings in a deck, which is 4. However, it is unclear how these concepts of microstate, macrostate, degeneracy, and multiplicity apply to a deck
teme92

## Homework Statement

Consider drawing one card from a deck with no jokers or other special cards.
a) What is the number of microstates? (4/13/52/cant tell)
b) What is the number of macrostates?(4/13/52/cant tell)
c) What is the degeneracy of macrostate spade? (4/13/52/cant tell)
d) What is the multiplicity of the macrostate King? (4/13/52/cant tell)

## The Attempt at a Solution

a) I said the number of microstates is equal to the number of cards types (Ace,2,3...K) so 13

b)Macrostates is equal to number of suits so 4

c)Dengeneracy of spades is equal to number of cards in spades so 13

d)I'm not sure but was thinking 4 because there are 4 kings in a deck

Could anyone please tell me if these are correct or if I am looking at it in the wrong way. Any help is much appreciated!

teme92 said:

## Homework Statement

Consider drawing one card from a deck with no jokers or other special cards.
a) What is the number of microstates? (4/13/52/cant tell)
b) What is the number of macrostates?(4/13/52/cant tell)
c) What is the degeneracy of macrostate spade? (4/13/52/cant tell)
d) What is the multiplicity of the macrostate King? (4/13/52/cant tell)
How are the terms microstate, macrostate, and degeneracy defined? These are not standard terms in probability.
teme92 said:

## The Attempt at a Solution

a) I said the number of microstates is equal to the number of cards types (Ace,2,3...K) so 13
Without knowing how your text defines micro/macrostate, I'd be inclined to think that there might be two macrostates: suit and rank. With the same thinking, there would be 52 microstates.
teme92 said:
b)Macrostates is equal to number of suits so 4

c)Dengeneracy of spades is equal to number of cards in spades so 13
How is degeneracy different from multiplicity?
teme92 said:
d)I'm not sure but was thinking 4 because there are 4 kings in a deck

Could anyone please tell me if these are correct or if I am looking at it in the wrong way. Any help is much appreciated!

Hey Mark44, thanks for the reply.

This is question for my Intro to Thermodynamics and Statistical Physics module but the problem is more general statistics.

A microstate is a specific microscopic configuration of a thermodynamic system that the system may occupy with a certain probability in the course of its thermal fluctuations. In contrast, the macrostate of a system refers to its macroscopic properties, such as its temperature and pressure.

In systems with multiple quantum states s sharing the same Es, it is said that the energy levels of the system are degenerate.

You said that there is two macrostates when you commented on part a) but in part b) there is no answer option for 2?

Mark44 said:
How are the terms microstate, macrostate, and degeneracy defined? These are not standard terms in probability.
Without knowing how your text defines micro/macrostate, I'd be inclined to think that there might be two macrostates: suit and rank. With the same thinking, there would be 52 microstates.
How is degeneracy different from multiplicity?

teme92 said:
Hey Mark44, thanks for the reply.

This is question for my Intro to Thermodynamics and Statistical Physics module but the problem is more general statistics.

A microstate is a specific microscopic configuration of a thermodynamic system that the system may occupy with a certain probability in the course of its thermal fluctuations. In contrast, the macrostate of a system refers to its macroscopic properties, such as its temperature and pressure.

In systems with multiple quantum states s sharing the same Es, it is said that the energy levels of the system are degenerate.

You said that there is two macrostates when you commented on part a) but in part b) there is no answer option for 2?

He did not say 2, he said 52.

Anyway, applying some of these concepts to a deck of cards seems like an inappropriate 'reach'. What are the energy levels of a card or a deck of cards?

Some of these concepts have differently-named versions in probability: "microstates" = sample space (at least in sufficiently fine-grained models) and "macrostates" = events in a partition of the sample space.

teme92 said:
Hey Mark44, thanks for the reply.

This is question for my Intro to Thermodynamics and Statistical Physics module but the problem is more general statistics.

A microstate is a specific microscopic configuration of a thermodynamic system that the system may occupy with a certain probability in the course of its thermal fluctuations. In contrast, the macrostate of a system refers to its macroscopic properties, such as its temperature and pressure.

In systems with multiple quantum states s sharing the same Es, it is said that the energy levels of the system are degenerate.

You said that there is two macrostates when you commented on part a) but in part b) there is no answer option for 2?
I said that I thought there might be two macrostates. Two macro properties of the cards would seem to me to be suit and rank. Questions c and d suggest that both suit and rank are macrostates. I can't explain why 2 isn't a possible answer, but I can't justify 4, 13, or 52, and "can't tell" doesn't seem to apply.

The module is using a deck of cards as an analog for a thermodynamic system, but I still don't understand how "degenerate" ties into cards. For question d, I believe the multiplicity of "King" is four, since there are four Kings, and I think that the multiplicity of "spade" is 13, since there are 13 spades, but how multiplicity and degeneracy are different, I don't understand.

If I were you I would contact the instructor to get his/her take on this.

Ok thanks guys for the help. I don't understand fully myself as there is no mention of this in any of the lectures. I'll just give him the answers I think are correct and tell him its his fault, for not being clear, if they are wrong :P

If these are questions he wrote himself, as opposed to ones that appear in the textbook, it could be that not all of them were as well thought-out as they should be. It has happened before...

No they aren't out of a textbook. They're on a sheet he handed out.

teme92 said:
No they aren't out of a textbook. They're on a sheet he handed out.
All the more reason to contact him, either in person or by email. You could tell him that you think the answers are (or should be) such and such, and give your reasoning.

teme92
Mark44 said:
I said that I thought there might be two macrostates. Two macro properties of the cards would seem to me to be suit and rank. Questions c and d suggest that both suit and rank are macrostates. I can't explain why 2 isn't a possible answer, but I can't justify 4, 13, or 52, and "can't tell" doesn't seem to apply.

I was thinking of part (a), where you did give the 52 answer.

The module is using a deck of cards as an analog for a thermodynamic system, but I still don't understand how "degenerate" ties into cards. For question d, I believe the multiplicity of "King" is four, since there are four Kings, and I think that the multiplicity of "spade" is 13, since there are 13 spades, but how multiplicity and degeneracy are different, I don't understand.

If I were you I would contact the instructor to get his/her take on this.

Ok, thanks again!

## 1. What is degeneracy in statistics?

Degeneracy in statistics refers to the occurrence of multiple solutions or outcomes for a given set of data. It can also refer to the presence of repeated or redundant data points in a dataset.

## 2. How does degeneracy affect statistical analysis?

Degeneracy can significantly impact statistical analysis by introducing bias and uncertainty into the results. It can also make it difficult to determine the most accurate or meaningful solution for a given problem.

## 3. What are some common causes of degeneracy in statistical data?

One common cause of degeneracy is the presence of multicollinearity, where two or more independent variables in a regression model are highly correlated. Other causes include measurement error, sampling bias, and the use of inappropriate statistical models.

## 4. How can degeneracy be addressed in statistical analysis?

There are various methods for addressing degeneracy, depending on the type and cause of the degeneracy. Some approaches include transforming the data, removing redundant variables, using regularization techniques, and implementing robust statistical methods.

## 5. What is multiplicity in statistics?

Multiplicity in statistics refers to the potential for multiple comparisons to be made within a single dataset or study. It can lead to an increased likelihood of finding false positive results if proper adjustments are not made to account for the increased number of comparisons.

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