We discuss that a single spin observable $\sigma_x$ in a quantum state does not have a counterpart in physical reality. We consider whether a single spin-1/2 pure state has a counterpart in physical reality. It is an eigenvector of Pauli observable $\sigma_z$ or an eigenvector of Pauli observable $\sigma_x$. We assume a state $|+_z\rangle$, which can be described as an eigenvector of Pauli observable $\sigma_z$. We assume also a state $|+_x\rangle$, which can be described as an eigenvector of Pauli observable $\sigma_x$. The value of transition probability $|\langle +_z|+_x\rangle|^2$ is 1/2. Surprisingly, the existence of a single classical probability space for the transition probability within the formalism of von Neumann's projective measurement does not coexist with the value of the transition probability $|\langle +_z|+_x\rangle|^2=1/2$. We have to give up the existence of such a classical probability space for the state $|+_z\rangle$ or for the state $|+_x\rangle$, as they define the transition probability. It turns out that the single spin-1/2 pure state $|+_z\rangle$ or the single spin-1/2 pure state $|+_x\rangle$ does not have counterparts in physical reality. A single spin-1/2 pure state (e.g., $|+ \rangle\langle +|$) is a single one-dimensional projection operator. In other word, a single one-dimensional projector does not have a counterpart in physical reality, in general.