Definition: We say that a subset $A\subset \mathbb{N}_0\times \mathbb{N}_0$ is symmetric if it satisfies the property: $(a,b)\in A \Rightarrow (b,a)\in A.$ We say that the subset $A$ is non-symmetric if it does not satisfies the property.
My question is the following
Question: Let $p\in (0,\frac{1}{2}).$ Is there any non-symmetric subset $A\subset \mathbb{N}_0\times \mathbb{N}_0$ such that $$\sum_{(a,b)\in A} p^a (1-p)^b=\sum_{(a,b)\in A} p^b (1-p)^a \mbox{ ?}$$