Take your qubit state $|\psi\rangle$ and let $|i\rangle$ some element in $\mathbb{C}^{2}$, say $|0\rangle$ or $|1\rangle$ etc $\{|u\rangle, |d\rangle\}$ since this probability space only has two state. We thus have
$$\langle i| \psi\rangle = \frac{1}{\sqrt{2}}\big\{\langle i|u\rangle - \langle i| d\rangle\big\}.$$
Now,
$$P(i):= |\langle i| \psi\rangle|^{2} = \frac{1}{2}\big\{|\langle i|u\rangle|^{2} + +|\langle i| d\rangle\big\}|^{2}+i\langle i|u\rangle\langle i| d\rangle^{*}-i\langle i|u\rangle^{*}\langle i| d\rangle.$$$$P(i):= |\langle i| \psi\rangle|^{2} = \frac{1}{2}\big\{|\langle i|u\rangle|^{2} +|\langle i| d\rangle\big\}|^{2}+i\langle i|u\rangle\langle i| d\rangle^{*}-i\langle i|u\rangle^{*}\langle i| d\rangle.$$