📝 Probability & Statistics

Conditional probability

$$P(y=1 \mid x)$$

means the probability that the outcome $y$ equals 1, given that we know the input features $x$.

The vertical bar $\mid$ means “given” or “conditioned on.”

Example

Suppose we want to predict whether a user will book a listing:

• $y=1$: the user books
• $y=0$: the user does not book
• $x$: information about the user, listing, and trip

For example:

$$x = [\text{price}, \text{location}, \text{rating}, \text{trip dates}, \text{user history}]$$

If the model estimates:

$$P(y=1 \mid x)=0.08$$

it means that, given these user, listing, and trip features, the estimated probability of booking is 8%.

Because this is binary classification:

$$P(y=0 \mid x)=1-P(y=1 \mid x)$$

Therefore:

$$P(y=0 \mid x)=1-0.08=0.92$$

In logistic regression

A logistic-regression model calculates:

$$z=\theta^\top x+\beta$$

and transforms it into a probability using the sigmoid function:

$$P(y=1 \mid x) =\sigma(z) =\frac{1}{1+e^{-z}}$$

For example, if $z=1.5$:

$$P(y=1 \mid x) =\frac{1}{1+e^{-1.5}} \approx 0.818$$

The model predicts an approximately 81.8% probability that $y=1$.

Important interpretation

This probability applies to observations with features like $x$; it does not guarantee what will happen in one individual case.

If:

$$P(y=1 \mid x)=0.8$$

a well-calibrated model should produce positive outcomes for approximately 80% of many similar examples. One specific example can still have $y=0$.