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In a ($2$ Player)game there are $2$ moves: $A,B$. The game is played $N$ times. The probability of choosing them is equal i.e. $0.5$ What is the probability that game will be won at least $M$ times by first player given first player moves first ?

I found out the condition of winning of each player without taking into account the probability of choosing the moves.

How can I think of the solution? What more do I need to calculate to find the probability. Can I calculate the probability in terms of given variables only?

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    What is the condition of victory? – 2017-02-10
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    @Wolfram The victory is based on nim game. Lets say the condition is if $X$ is even then it's a victory for first player else second player wins. – 2017-02-10
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    What is $X$? What do moves $A$ and $B$ do? I just don't understand the nature of the game, do you mean that $A$ or $B$ are the moves that players do independently one time in a game? – 2017-02-10
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    $A$ and $B$ are two different set of moves. A player can choose either $A$ or $B$. Now I created a variable $X$ which tells me whether first player wins or second which is independent of set of moves. – 2017-02-10

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