Assume that we model a game in tennis as follows: if player A serves, they have a probability of holding their service (win a point) pA and player B, a probability pB.
The idea is assuming A serves:
p(0,0) = 1 (before match starts)
p(1,0) = pA * p(0,0)
p(0,1) = (1-pA) * p(0,0)
p(1,1) = pA * p(0,1) + (1-pA) * p(1,0)
p(2,0) = pA * p(1,0)
etc
same notation for player B.
Here is my question to you: I have two methodologies for figuring out the probability of player A or B winning the game if we don't know who serves (the derive different results) and I am not sure which one is correct:
Method 1
I derive my probabilities assuming player A serves -> table 1 I derive my probabilities assuming player B server -> table 2
my desirable table of probs is the average (table1 + table 2)/2 and player A winning is = p(4,0) + p(4,1) + p(4,2) + p(win by advantage)
Method 2
Player A has an probability pA if servers and (1-pB) if receives, hence
overall probability (pA + (1-pB))/2 = pC
and now I model a game using pC
These two methods return different results
Example
Method 1 pA = 0.65 pB = 0.60
A winning 4-0 as server : p(4,0) = pA^4 = 0.65^4 = 0.1758 A winning 4-0 as receiver: p(4,0) = (1-pB)^4 = 0.0256
A winning 4-0 on average: 0.1020
Method 2
pC = (0.65 + (1-0.60))/2 = 0.525
A winning 4-0 on average: p(4,0) = 0.525 ^ 4 = 0.0757