Question about a property of a matrix of transition probabilities

In summary, the conversation discusses the mathematical concepts and probability distributions related to golf hole scores, specifically on par three holes. The article mentions transition probabilities and the use of a matrix to calculate probabilities after multiple steps. The conversation also mentions references and the possibility of finding more information in textbooks using formal mathematical terms.
  • #1
Ad VanderVen
169
13
In a 2012 article published in the Mathematical Gazette, in the game of golf hole score probability distributions were derived for a par three, four and five based on Hardy's ideas of how an hole score comes about. Hardy (1945) assumed that there are three types of strokes: a good (##G##) stroke, a bad (##B##) stroke and an ordinary (##O##) stroke, where the probability of a good stroke equals ##p##, the probability of a bad stroke equals ##q## and the probability of a ordinary stroke equals ##1 - p - q##. In fact, Hardy called a good shot a super shot and a bad shot a sub shot. Minton (2010) later called Hardy's super shot an excellent shot (##E##) and Hardy's sub shot a bad shot (##B##). Here Minton's excellent shot is called a good shot (##G##). Hardy further assigned a value of 2 to a good stroke, a value of 0 to a bad stroke and a value of 1 to a regular or ordinary stroke. Once the sum of the values is greater than or equal to the value of the par of the hole, the number of strokes in question is equal to the score obtained on that hole. In the 2012 article, the probability distribution of hole score ##X## on a par three is written as ##P(X_{3}=k)## for ##k = 2, 3, \dots## . To find a general expression for the probability ##P(X_{3}=k)## for ##k = 2, 3, \dots##, the following matrix of transition probabilities was given.

01234
0q1-p-qp00
10q1-p-qp0
200q1-p-qp
300010
400001

Here this matrix is referred to as ##M_3##. According to the definition of the matrix of transition probabilities ##M_{3}## and the property, that ##M_{3}^ k## gives the transition probabilities after ##k## steps, one may write

##P (X_{3} = k) = (M_{3}^{k} - M_{3}^{k-1})_{1,4} + (M_{3}^{k} - M_{3}^{k-1})_{1,5}##

where ##M_3## refers to the matrix in van der Ven (2012).

Now my question is whether this last property is also mentioned somewhere in probability theory textbooks in which transition probability matrices are discussed.

References

Hardy, G.H. (1945). A mathematical theorem about golf. The Mathematical Gazette, 29, pp. 226 - 227.
Ven, A.H.G.S. van der (2012). The Hardy distribution for golf hole scores. The Mathematical Gazette, 96, pp. 428 - 438.
Minton, R.B. (2010). G. H. Hardy's Golfing Adventure, Mathematics and sports, Joseph A. Gallian, ed. MAA pp. 169-179.
 
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  • #3
FactChecker said:
In the article you can read:

On a par three hole, for example, there are transition states 0, 1, 2 corresponding to the result of an initial bad, ordinary or good shot, respectively, and there are two absorption states 3 and 4 corresponding to holing out. More generally, on a par ##N## hole the states of the system are ##0, 1, ..., N-1##, and the transitions between the states are governed by the following rule: once the player reaches state ##N## or ##N+1##, no further transition into another state is possible; when the player is at state ##k##, with ##0 <= k<= N-1##, then the next transition is either to the same state with probability ##q##, or to ##k+1## with probability (##1-p-q##), or to ##k+2## with probability ##p##. This type of system is called a random walk with absorbing barriers at states ##N## and ##N+1##.
 

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  • #4
A textbook is likely to use formal mathematical terms. I do not see the reference list in the article pages that you attached but there are some references made. Do you have the reference list and have you checked them?
CORRECTION: I see that the last page of the article, with the list of references, is actually the first page of the attachment.
 
  • #5
FactChecker said:
A textbook is likely to use formal mathematical terms. I do not see the reference list in the article pages that you attached but there are some references made. Do you have the reference list and have you checked them?
CORRECTION: I see that the last page of the article, with the list of references, is actually the first page of the attachment.
I do not care about "A textbook is likely to use formal mathematical terms." Any textbook would suffice.
 
  • #6
Ad VanderVen said:
I do not care about "A textbook is likely to use formal mathematical terms." Any textbook would suffice.
Sorry. I was not clear. In searching for more information in textbooks, it is more likely to find it by using the correct mathematical terms.

PS. It seems like what you want requires more expertise than I have. I will stay out of this and let people who know more about it respond.
 
Last edited:
  • #7
FactChecker said:
Sorry. I was not clear. In searching for more information in textbooks, it is more likely to find it by using the correct mathematical terms.

PS. It seems like what you want requires more expertise than I have. I will stay out of this and let people who know more about it respond.
Thanks for your responses anyway.
 

1. What is a matrix of transition probabilities?

A matrix of transition probabilities is a mathematical tool used in probability theory and statistics to represent the likelihood of transitioning from one state to another. It is commonly used in Markov chain analysis to model the behavior of a system over time.

2. How is a matrix of transition probabilities calculated?

A matrix of transition probabilities is calculated by dividing the number of transitions from one state to another by the total number of transitions from that state. This process is repeated for each state, resulting in a matrix with probabilities as its elements.

3. What is the significance of a matrix of transition probabilities?

A matrix of transition probabilities is significant because it allows us to make predictions about the future behavior of a system based on its current state. It is also useful for analyzing the long-term behavior of a system and identifying any steady-state probabilities.

4. Can a matrix of transition probabilities have negative values?

No, a matrix of transition probabilities cannot have negative values. The probabilities must be between 0 and 1, as they represent the likelihood of transitioning from one state to another. Negative values would not make sense in this context.

5. How is a matrix of transition probabilities used in real-world applications?

A matrix of transition probabilities is used in a variety of real-world applications, such as in finance to model stock market behavior, in biology to study ecological systems, and in computer science to analyze network traffic. It is also commonly used in machine learning and data analysis to make predictions and identify patterns in data.

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