JuniorTheoryCommon

What does this program print?

Python

import numpy as np

w, b = 2.0, -3.0
x = np.array([1.5, 3.0])
z = w * x + b
p = 1 / (1 + np.exp(-z))
print(round(p[0], 2), round(p[1], 2))

0.5 0.95

Expecting 0.0 for a score of zero: the sigmoid of 0 is 1 / (1 + 1) = 0.5, the point where the model sees both classes as equally likely.
Expecting 0.05 for a score of 3: with np.exp(-z) in the denominator a positive score gives a probability above 0.5; 0.05 would need np.exp(z).

Which value of x gives a probability of exactly 0.5 when w = 2.0 and b = -3.0?
Why can the output of the sigmoid be read as a probability for any score?

JuniorTheoryCommon

Python

import numpy as np

w, b = 2.0, -3.0
x = np.array([1.5, 3.0])
z = w * x + b
p = 1 / (1 + np.exp(-z))
print(round(p[0], 2), round(p[1], 2))

0.5 0.95

Expecting 0.0 for a score of zero: the sigmoid of 0 is 1 / (1 + 1) = 0.5, the point where the model sees both classes as equally likely.
Expecting 0.05 for a score of 3: with np.exp(-z) in the denominator a positive score gives a probability above 0.5; 0.05 would need np.exp(z).

Which value of x gives a probability of exactly 0.5 when w = 2.0 and b = -3.0?
Why can the output of the sigmoid be read as a probability for any score?