Bayes with Total Probability Practice Problem
This data science coding problem helps you practice Conditional Probability & Bayes, bayes with total probability, and implementation skills. Read the problem statement, write your solution, and strengthen your understanding of Conditional Probability & Bayes.
- Problem ID: 234
- Problem key: 234-bayes-with-total-probability
- URL: https://datacrack.app/solve/234-bayes-with-total-probability
- Difficulty: medium
- Topic: Conditional Probability & Bayes
- Module: Probability
Problem Statement
# 🧩 Bayes with Total Probability
---
### 🎯 Goal
Use total probability to calculate the evidence, then use Bayes’ theorem to calculate a posterior probability.
---
### 📖 Introduction
In the previous problem, Bayes’ theorem used evidence that was already given:
$$
P(B)
$$
But often $P(B)$ is not given directly.
Instead, you may know:
$$
P(A)
$$
$$
P(B\mid A)
$$
and:
$$
P(B\mid A^c)
$$
Then you first calculate the evidence using total probability:
$$
P(B)=P(B\mid A)P(A)+P(B\mid A^c)P(A^c)
$$
After that, use Bayes’ theorem:
$$
P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}
$$
This problem combines the previous ideas into one full Bayes calculation.
---
### 📥 Input
- `p_a`: prior probability $P(A)$.
- `p_b_given_a`: likelihood $P(B\mid A)$.
- `p_b_given_not_a`: probability $P(B\mid A^c)$.
### 📤 Output
- Return $P(A\mid B)$ rounded to 6 decimals.
---
### 💻 Task
Implement `bayes_with_total_probability(p_a, p_b_given_a, p_b_given_not_a)`.
Use the provided probabilities to calculate the posterior:
$$
P(A\mid B)
$$
---
### 🧩 Starter Code
```python
def bayes_with_total_probability(p_a, p_b_given_a, p_b_given_not_a):
pass
```
---
### 💡 Example
```python
bayes_with_total_probability(0.3, 0.8, 0.2)
```
The function should return the posterior probability as a decimal rounded to 6 decimals.
---
### ⚠️ Common Mistakes
- Using Bayes’ theorem before calculating $P(B)$.
- Forgetting to calculate $P(A^c)$.
- Using only the $A$ path and ignoring the $A^c$ path.
- Confusing $P(B\mid A^c)$ with $P(A^c\mid B)$.
Starter Code
def bayes_with_total_probability(p_a, p_b_given_a, p_b_given_not_a):
pass