A score has no natural limit
A linear model produces a score z = wx + b. That score can be negative or larger than one, so it is not itself a probability. The sigmoid function maps any real score to a number between zero and one.
Separate the probability from the decision
Move the score below and watch the point travel along the sigmoid curve. Then move only the threshold: the dashed line moves, and the predicted class can change even though the model probability stays the same. At score zero, the sigmoid gives 0.5. First move the score; then leave the score alone and move the threshold. The curve gives the probability, while the threshold decides the label.
INTERACTIVE EXPLORATION
From score to a decision
Move the score and the threshold independently. A probability does not change when you change only the decision threshold.
What does the number mean?
Logistic regression is commonly trained with binary log loss. A sigmoid output is a probability-shaped number; whether it is well calibrated must be checked on suitable data. The threshold is a decision rule and can depend on the costs of different errors.
Check your understanding
With z = 0 and threshold 0.5, what class is predicted if ties count as positive?
The sigmoid is 0.5, so the positive class is predicted under this tie rule.