Explanation

In logistic regression, the probability that the dependent variable equals 1 is given by the logistic function:

$P(y_i = 1) = \frac{1}{1 + e^{-(\alpha + \beta_1 x_{1i} + \beta_2 x_{2i} + \beta_3 x_{3i})}} = \frac{1}{1 + e^{-y_i}}$

Since this is a binary classification problem where $y_i$ can only be 0 or 1, the probability that $y_i = 0$ is the complement:

$P(y_i = 0) = 1 - P(y_i = 1) = 1 - \frac{1}{1 + e^{-y_i}} = \frac{e^{-y_i}}{1 + e^{-y_i}} = \frac{1}{1 + e^{y_i}}$

Therefore, the correct answer is $\frac{1}{1 + e^{y_i}}$.

Key Points:

• Logistic regression transforms linear combinations into probabilities using the logistic function
• The probability of class 0 is the complement of the probability of class 1
• The transformation ensures probabilities are bounded between 0 and 1