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In a game a coin is tossed (2n+m) times and a player wins if he does not get any two consecutive outcomes same for at least 2n times in a row. How to calculate probability that player wins game.

I can understand the outcomes will be (HTHTHTHTHTHTH...) OR (THTHTHTHTHTHTHTH...) atleast 2n times but how to relate it to get probability ?

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    Not sure I understand the rules. Do you mean that there are at least $2n$ throws which do not match their predecessor? Does the first throw count? Let's do $n=1,m=1$ explicitly. Then the possible outcomes are $HHH,HHT,HTH,THH,HTT,THT,TTH,TTT$. Which of these are winners? – 2017-02-22
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    Supposing that, for the $n=1,m=1$ case, you say that only $THT$ and $HTH$ are winners then: Except for the first throw, every toss has a well defined predecessor. The probability that this toss doesn't match the predecessor is $\frac 12$. Thus you have $2n+m-1$ independent Bernoulli trials and you demand at least $2n$ wins. Easy computation. – 2017-02-22

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$2^{2n+m}$ outcomes in total.

desired ways are $\dbinom{2n+m}{2n}\times 2$

so, probability is

$\dfrac{\dbinom{2n+m}{2n}\times 2}{2^{2n+m}}$

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    didnt get how you got desired ways..its atleast 2n not exaxctly 2n ?? – 2017-02-22