The conditional probability that event A occurs, given that event B has occurred, is calculated as follows:
P(A|B) = P(A∩B) / P(B)
where:
P(A∩B) = the probability that event A and event B both occur.
P(B) = the probability that event B occurs.
The following example shows how to use this formula to calculate conditional probabilities in Python.
Example: Calculate Conditional Probability in Python
Suppose we send out a survey to 300 individuals asking them which sport they like best: baseball, basketball, football, or soccer.
We can create the following table in Python to hold the survey responses:
import pandas as pd
import numpy as np
#create pandas DataFrame with raw data
df = pd.DataFrame({'gender': np.repeat(np.array(['Male', 'Female']), 150),
'sport': np.repeat(np.array(['Baseball', 'Basketball', 'Football',
'Soccer', 'Baseball', 'Basketball',
'Football', 'Soccer']),
(34, 40, 58, 18, 34, 52, 20, 44))})
#produce contingency table to summarize raw data
survey_data = pd.crosstab(index=df['gender'], columns=df['sport'], margins=True)
#view contingency table
survey_data
sport Baseball Basketball Football Soccer All
gender
Female 34 52 20 44 150
Male 34 40 58 18 150
All 68 92 78 62 300
Related: How to Use pd.crosstab() to Create Contingency Tables in Python
We can use the following syntax to extract values from the table:
#extract value in second row and first column
survey_data.iloc[1, 0]
[1] 34
We can use the following syntax to calculate the probability that an individual is male, given that they prefer baseball as their favorite sport:
#calculate probability of being male, given that individual prefers baseball
survey_data.iloc[1, 0] / survey_data.iloc[2, 0]
0.5
And we can use the following syntax to calculate the probability that an individual prefers basketball as their favorite sport, given that they’re female:
#calculate probability of preferring basketball, given that individual is female
survey_data.iloc[0, 1] / survey_data.iloc[0, 4]
0.3466666666666667
We can use this basic approach to calculate any conditional probability we’d like from the contingency table.
Additional Resources
The following tutorials provide additional information on dealing with probability:
Law of Total Probability
How to Find the Mean of a Probability Distribution
How to Find the Standard Deviation of a Probability Distribution