Graduate Question about a property of a matrix of transition probabilities

[Ad VanderVen]

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.

	0	1	2	3	4
0	q	1-p-q	p	0	0
1	0	q	1-p-q	p	0
2	0	0	q	1-p-q	p
3	0	0	0	1	0
4	0	0	0	0	1

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.

 

[FactChecker]

Is this the same as the k-step transition probability of a discrete-time Markov chain?

 

[Ad VanderVen]

Is this the same as the k-step transition probability of a discrete-time Markov chain?

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 $N+1$.

 

[FactChecker]

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.

 

[Ad VanderVen]

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.

 

[FactChecker]

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.

 

[Ad VanderVen]

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.