This paper introduces the method of symmetrization of the measure algebra of a compact $P_ * $-hypergroup. This method is used to form a measure algebra whose characters are Krawtchouk polynomials (these are the finite sets of polynomials orthogonal with respect to the binomial distribution on $\{ 0,1, \cdots ,N\} $. As a further application, one derives a theorem about nonnegative expansions of one family of Krawtchouk polynomials in terms of another family.