The probability that a line through the origin pierces the cone genera
ted by n vectors in R(m) chosen with a centrally symmetric distributio
n is calculated. This is used to derive the probability of the existen
ce of a Nash equilibrium for a random pay-off matrix A, for which p(i)
is never zero, or is zero for only one component, given some assumpti
ons about the probability distribution of the elements of A.