Why we need sigmoid

Pass/fail labels are 0 or 1. Raw neuron output can be -3, 0.7, or 12. We want something between 0 and 1 we can read as “chance of pass.”

Sigmoid does that:

import math

def sigmoid(x):
    return 1 / (1 + math.exp(-x))

Raw output (z)	Sigmoid	Plain read
-2.0	~0.12	Likely fail
0.0	0.50	Coin flip
1.2	~0.77	Likely pass
4.0	~0.98	Very likely pass

Raw score → probability → label

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Frames:

Raw z can be any real number—too wild for a probability.
Sigmoid bends z into the range (0, 1).
We pick 0.5 as the cutoff: at or above → Pass, below → Fail.

Make a prediction

raw_output = 1.2
probability = sigmoid(raw_output)
print(probability)  # ~0.7685

if probability >= 0.5:
    print("Pass")
else:
    print("Fail")

The model is not thinking. It applies a formula, then compares to a line you chose (0.5). Different problems use different thresholds; 0.5 is the usual teaching default.

Full forward pass in one snippet


              import math

def sigmoid(x):
  return 1 / (1 + math.exp(-x))

study_hours = 3
weight = 0.5
bias = -1

raw = study_hours * weight + bias
prob = sigmoid(raw)

print("raw:", raw)
print("probability:", round(prob, 4))
print("label:", "Pass" if prob >= 0.5 else "Fail")


              import math

def sigmoid(x):
  return 1 / (1 + math.exp(-x))

def predict_pass(study_hours, weight, bias):
  raw = study_hours * weight + bias
  prob = sigmoid(raw)
  label = "Pass" if prob >= 0.5 else "Fail"
  return label, prob

print(predict_pass(2, 0.5, -1))
print(predict_pass(3, 0.5, -1))

Live runner — tweak hours

🐍 Python Sigmoid + threshold

📟 Console Output

Run code to see output...

Exercise 2 — uncertain zone

Run with study_hours = 2.5, weight = 0.5, bias = -1.

Question: Is the probability close to 0.5? Why does that feel like “unsure”?

Answer

raw = 2.5×0.5 - 1 = 0.25, sigmoid(0.25) ≈ 0.56—just above 0.5. Small changes in hours or weights could flip the label. That is uncertainty, not the model “being shy.”

Key takeaways

Sigmoid maps any real number into (0, 1).
Threshold 0.5 turns probability into Pass/Fail.
Predictions are math + a rule—not reasoning.

Next: when the prediction is wrong, we measure loss and start training.

Progress 43%

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