# Conditional Probability in Python

1. Theory behind conditional probability
2. Example with python

# Part 1: Theory and formula behind conditional probability

For once, wikipedia has an approachable definition,

In probability theoryconditional probability is a measure of the probability of an event occurring given that another event has (by assumption, presumption, assertion or evidence) occurred.

Translation: given B is true, what is the probability that A is also true.

It’s easier to understand something with concrete examples. Below are a few random examples of conditional probabilities we could calculate.

## Examples:

1. What’s the probability of someone sleeping less than 8 hours if they’re a college student.
2. What’s the probability of a dog living longer than 15 years if they’re a border collie.
3. What’s the probability of using all your vacation days if you work for the government.

## Formula:

The formula for conditional probability is `P(A|B) = P(A ∩ B) / P(B)`.

The parts:
P(A|B) = probability of A occurring, given B occurs
P(A ∩ B) = probability of both A and B occurring
P(B) = probability of B occurring

`|` means “given”. Meaning “in cases where something else occurs”.

`∩` means intersection which you can think of as `and`, or the overlap in the context of a Venn diagram.

But why do we divide `P(A ∩ B)` by `P(B)`in the formula?

Because we want to exclude the probability of non-B cases. We’re scoping our probability to that falling within `B`.

Dividing by`P(B)` removes the probability of anything not `B` . `C — B` above.

# Part 2: Example with python

We’re going to calculate the probability a student gets an A (80%+) in math, given they miss 10 or more classes.

`import pandas as pddf = pd.read_csv('student-alcohol-consumption/student-mat.csv')df.head(3)`

And check the number of records.

`len(df)#=> 395`

We’re only concerned with the columns, `absences` (number of absences), and `G3` (final grade from 0 to 20).

Let’s create a couple new boolean columns based on these columns to make our lives easier.

Add a boolean column called `grade_A` noting if a student achieved 80% or higher as a final score. Original values are on a 0–20 scale so we multiply by 5.

`df['grade_A'] = np.where(df['G3']*5 >= 80, 1, 0)`

Make another boolean column called `high_absenses` with a value of 1 if a student missed 10 or more classes.

`df['high_absenses'] = np.where(df['absences'] >= 10, 1, 0)`

Add one more column to make building a pivot table easier.

`df['count'] = 1`

And drop all columns we don’t care about.

`df = df[['grade_A','high_absenses','count']]df.head()`

Nice. Now we’ll create a pivot table from this.

`pd.pivot_table(    df,     values='count',     index=['grade_A'],     columns=['high_absenses'],     aggfunc=np.size,     fill_value=0)`

We now have all the data we need to do our calculation. Let’s start by calculating each individual part in the formula.

In our case:
`P(A)` is the probability of a grade of 80% or greater.
`P(B)` is the probability of missing 10 or more classes.
`P(A|B)` is the probability of a 80%+ grade, given missing 10 or more classes.

Calculations of parts:
P(A) = (35 + 5) / (35 + 5 + 277 + 78) = 0.10126582278481013
P(B) = (78 + 5) / (35 + 5 + 277 + 78) = 0.21012658227848102
P(A ∩ B) = 5 / (35 + 5 + 277 + 78) = 0.012658227848101266

And per the formula, `P(A|B) = P(A ∩ B) / P(B)`, put it together.

P(A|B) = 0.012658227848101266/ 0.21012658227848102= 0.06

There we have it. The probability of getting at least an 80% final grade, given missing 10 or more classes is 6%.