X is number of (fair) coin flips needed to land m heads OR m tails. m is arbitrary natural number. Delfine CDF of X. (in It's simplest form)

What I already know:

  • $P(x=k) = {{k-1} \choose{m-1}}*(1/2)^{k-1}$

    I suspect one way to get CDF is to Sum P(x=k) from k=m to x, and Mathematica evaluates that sum as:$ 2-2^{x}{{x} \choose{m-1}}*hypergeometric(1,x+1;x-m+2;1/2) $
    But I would like to know how to get to that without computer. Or is there another way to get to CDF of X? 


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