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. The probability that player wins the game is

Player should get (HT, HT, HT,...) or (TH, TH,...) at least 2n times. If the sequence starts from first place, then the probability is 1/22n and if starts from any other place, then the probability is 122n+1 Hence, required probability is2122n+m22n+1=m+222n