Classical probability assigns nonnegative probabilities directly to possible outcomes. Quantum theory instead represents a system by a quantum state containing complex probability amplitudes.

For mutually exclusive alternatives labeled by $i$, write amplitudes

$$\psi_i\in\mathbb C.$$

The probability of outcome $i$ is obtained from the squared magnitude:

$$P(i)=|\psi_i|^2.$$

For a normalized state,

$$\sum_i|\psi_i|^2=1.$$

Why amplitudes are different from probabilities

Amplitudes can have phase. Before probabilities are calculated, amplitudes associated with indistinguishable alternatives can add:

$$\psi=\psi_1+\psi_2.$$

Then

$$|\psi|^2$$

contains an interference term that would not appear if ordinary probabilities were simply added.

Superposition

If $|a\rangle$ and $|b\rangle$ are possible state vectors, then a normalized combination

$$|\psi\rangle=\alpha|a\rangle+\beta|b\rangle$$

is also a possible quantum state when

$$|\alpha|^2+|\beta|^2=1.$$

This superposition principle is a structural feature of quantum states, not a claim that a classical object literally possesses several definite values at once.

Complex amplitudes provide the mathematical layer from which observable probabilities and interference emerge.