In Linear Regression we multiply the coefficients of our features by their respective feature values and add the intercept, resulting in our prediction, which can range from -∞ to +∞. In Logistic Regression, we make the same multiplication of feature coefficients and feature values and add the intercept, but instead of the prediction, we get what is called the * log-odds*.

The log-odds are another way of expressing the probability of a sample belonging to the positive class, or a student passing the exam. In probability, we calculate the *odds* of an event occurring as follows:

`$Odds = \frac{P(event\ occurring)}{P(event\ not\ occurring)}$`

The odds tell us how many more times likely an event is to occur than not occur. If a student will pass the exam with probability `0.7`

, they will fail with probability `1 - 0.7 = 0.3`

. We can then calculate the odds of passing as:

`$Odds\ of\ passing = \frac{0.7}{0.3} = 2.\overline{33}$`

The log-odds are then understood as the logarithm of the odds!

`$Log\ odds\ of\ passing = log(2.\overline{33}) = 0.847$`

For our Logistic Regression model, however, we calculate the log-odds, represented by `z`

below, by summing the product of each feature value by its respective coefficient and adding the intercept. This allows us to map our feature values to a measure of how likely it is that a data sample belongs to the positive class.

`$z = b_{0}+b_{1}x_{1} + \cdots + b_{n}x_{n}$`

`b_0`

is the intercept`b_1`

,`b_2`

, …`b_n`

are the coefficients of the features`x_1`

,`x_2`

, …`x_n`

This kind of multiplication and summing is known as a *dot product*.

We can perform a dot product using `numpy`

‘s `np.dot()`

method! Given feature matrix `features`

, coefficient vector `coefficients`

, and an `intercept`

, we can calculate the log-odds in `numpy`

as follows:

log_odds = np.dot(features, coefficients) + intercept

`np.dot()`

will take each row, or student, in `features`

and multiply each individual feature value by its respective coefficient in `coefficients`

, summing the result, as shown below.

We then add in the intercept to get the log-odds!

### Instructions

**1.**

Let’s create a function `log_odds`

that takes `features`

, `coefficients`

and `intercept`

as parameters. For now return `features`

.

**2.**

Update `log_odds`

to return the dot product of `features`

and `coefficients`

.

**3.**

Update the `return`

statement of log-odds by adding the `intercept`

after the dot product.

**4.**

With the `log_odds`

function you created, let’s calculate the log-odds of passing for the Introductory Machine Learning students. Use `hours_studied`

as the features, `calculated_coefficients`

as the coefficients and `intercept`

as the intercept. Store the result in `calculated_log_odds`

, and print it out.

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