Integral representations for binomial sums of chances of winning

Addona, Wagon and Wilf ( Ars Math. Contemp. 4 (1) (2011), 29-62) examined a problem about the winning chances in tossing unbalanced coins. Here we present some integral representations associated with such winning probabilities in a more general setting via using certain Fourier transform method. When our newly introduced parameters ( r ,  d ) are set to be (0, 1) , one of our results reduces to the main formula in the above reference.