Probit- and Logit for Winning Probabilities

#1
Hello guys, Ive got an issue with a log. regression and I think its mostly a logical mistake, maybe you can help me out.

What I want to do:
1. Use a binary model, use different variables to predict 0 = Lose the game, 1= win the game.
2. Check the regression data the probit / logit and LS regression gives you.
3. Use them and get the probabilities for winning (=1) and losing (=0).

Here is my issue (and I think I got a lot more, unsolved yet as I don't know them actually).
I got the y=0 values and y=1 values in the regression.
Therefore I get probabilities for the winner and the loser of the match. Those summed probabilities are >1.

Any idea how to deal with this? The mistake has to be already in the first 3 steps. Usually I should be able to work with p and 1-p.
But as 0 and 1 values have to be in the model, I cant just deal with winners probabilities, can I?

I will attach the results and my excelsheet in which I tried to sum it all up (not the complete sheet).


1563196093018.png 1563196105839.png 1563196108766.png
 

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#4
For head to head comparisons with a winner a fairly simple approach is a Bradley-Terry model: https://en.wikipedia.org/wiki/Bradley–Terry_model
The approach would generally work, however I wonder, if I am working with the Logistic Prob. Models in a correct way.
Thanks for your help and Input @Dason !

Is there any way to NOT use 0 and 1 only on logit or probit approaches?
Like, could I set the actual predicted probability (consider it known) as =y and then predict my own due to my variables in some way?
 
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