So far, we’ve learned that the equation for a logistic regression model looks like this:

`$ln(\frac{p}{1-p}) = b_{0} + b_{1}x_{1} + b_{2}x_{2} +\cdots + b_{n}x_{n}$`

Note that we’ve replaced *y* with the letter *p* because we are going to interpret it as a probability (eg., the probability of a student passing the exam). The whole left-hand side of this equation is called * log-odds* because it is the natural logarithm (

*ln*) of odds (

*p/(1-p)*). The right-hand side of this equation looks exactly like regular linear regression!

In order to understand how this link function works, let’s dig into the interpretation of * log-odds* a little more. The

*odds*of an event occurring is:

`$Odds = \frac{p}{1-p} = \frac{P(event\ occurring)}{P(event\ not\ occurring)}$`

For example, suppose that the probability a student passes an exam is *0.7*. That means the probability of failing is *1 - 0.7 = 0.3*. Thus, the odds of passing are:

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

This means that students are 2.33 times more likely to pass than to fail.

Odds can only be a positive number. When we take the natural log of odds (the log odds), we transform the odds from a positive value to a number between negative and positive infinity — which is exactly what we need! The logit function (log odds) transforms a probability (which is a number between 0 and 1) into a continuous value that can be positive or negative.

### Instructions

**1.**

Suppose that there is a 40% probability of rain today (p = 0.4). Calculate the odds of rain and save it as `odds_of_rain`

. Note that the odds are less than 1 because the probability of rain is less than 0.5.

**2.**

Use the odds that you calculated above to calculate the log odds of rain and save it as `log_odds_of_rain`

. You can calculate the natural log of a value using the `numpy.log()`

function. Note that the log odds are negative because the probability of rain was less than 0.5.

**3.**

Suppose that there is a 90% probability that my train to work arrives on-time. Calculate the odds of my train being on-time and save it as `odds_on_time`

. Note that the odds are greater than 1 because the probability is greater than 0.5.

**4.**

Use the odds that you calculated above to calculate the log odds of an on-time train and save it as `log_odds_on_time`

. Note that the log odds are positive because the probability of an on-time train was greater than 0.5.