For an infinite Bernoulli convolution whose support is a symmetric Cantor-type set, R. Kershner and A. Wintner [14] proved that the convolution is absolutely continuous or singular depending on whether the Lebesgue measure of the support is positive or zero respectively. Their proof employs the same construction which was developed originally by F. Hausdorff [11] in his celebrated paper, ‘Dimension und äusseres Mass’, where he introduced the notion of a fractional dimension. In this article we exploit this observation and prove a result which extends the theorem of Kershner and Wintner by considering Hausdorff instead of Lebesgue measure for the same class of infinite Bernoulli convolutions.