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Analyze a Tennis Game

In a game of tennis between two players, suppose the probability of the server winning a point is . Visualize the random walk graph for , with 17 states.

    
Wolfram Language code: proc = With[{p = 3 / 5, q = 2 / 5}, DiscreteMarkovProcess[17, (⁠| | | | | | | | | | | | | | | | | | | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | | 0 | 0 | 0 | p | q | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | | p | 0 | q | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | | 0 | q | p | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | | p | 0 | 0 | q | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | | 0 | q | 0 | 0 | p | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | | p | 0 | 0 | 0 | 0 | q | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | | 0 | 0 | q | 0 | 0 | p | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | | 0 | 0 | p | 0 | 0 | 0 | q | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | | 0 | q | 0 | 0 | 0 | 0 | p | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | | 0 | 0 | 0 | 0 | 0 | 0 | 0 | p | q | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | p | q | 0 | 0 | 0 | 0 | 0 | 0 | 0 | | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | p | q | 0 | 0 | 0 | 0 | 0 | 0 | | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | p | q | 0 | 0 | 0 | 0 | | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | p | q | 0 | 0 | 0 | | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | p | q | 0 |⁠)]];
Wolfram Language code: Graph[proc, VertexSize -> {"Scaled", 0.07}, VertexLabels -> Thread[Range[17] -> Map[Placed[Style[#, {Bold}], Center]&, states = {"GameA", "GameB", "Deuce", "AdvA", "AdvB", "40-15", "15-40", "40-0", "30-15", "15-30", "0-40", "30-0", "15-15", "0-30", "15-0", "0-15", "0-0"}]], ImageSize -> Large, ImagePadding -> 25]
Find the probability of the server winning the game for :
Wolfram Language code: PDF[proc[∞], 1]//N
Find the mean time to absorption, that is, the number of points played:
Wolfram Language code: N[Mean[FirstPassageTimeDistribution[proc, {1, 2}]]]
Find the mean number of states visited:
Wolfram Language code: MarkovProcessProperties[proc, "TransientTotalVisitMean"]//N
Find the average number of times the score will be tied at deuce:
Wolfram Language code: MarkovProcessProperties[proc, "TransientVisitMean"][[1]]//N