Sum One, Somewhere

For a fixed p, independently label the nodes of an infinite complete binary tree 0 with probability p, and 1 otherwise. For what p is there exactly a 1/2 probability that there exists an infinite path down the tree, starting at the root, that sums to at most 1 (that is, all nodes visited, with the possible exception of one, will be labeled 0). Find this value of p, either exactly or as a decimal accurate to 10 decimal places.