Learn Implementing Conditional Probability & Bayes' Theorem in Python

Conditional Probability

Conditional probability measures the chance of an event happening given another event has already occurred.

Formula:

P(A \mid B) = \frac{P(A \cap B)}{P(B)}


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P_A_and_B = 0.1  # Probability late AND raining
P_B = 0.2        # Probability raining

P_A_given_B = P_A_and_B / P_B
print(f"P(A|B) = {P_A_given_B:.2f}")  # Output: 0.5

Interpretation: if it is raining, there's a 50% chance you will be late to work.

Bayes' Theorem

Bayes' Theorem helps us find $P(A|B)$ when it's hard to measure directly, by relating it to $P(B|A)$ which is often easier to estimate.

Formula:

P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)}

Where:

$P(A \mid B)$ - probability of A given B (our goal);
$P(B \mid A)$ - probability of B given A;
$P(A)$ - prior probability of A;
$P(B)$ - total probability of B.

Expanding $P(B)$

P(B) = P(B \mid A) P(A) + P(B \mid \neg A) P(\neg A)


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P_A = 0.01                # Disease prevalence
P_not_A = 1 - P_A
P_B_given_A = 0.99        # Sensitivity
P_B_given_not_A = 0.05    # False positive rate

# Total probability of testing positive
P_B = (P_B_given_A * P_A) + (P_B_given_not_A * P_not_A)
print(f"P(B) = {P_B:.4f}")  # Output: 0.0594

# Apply Bayes’ Theorem
P_A_given_B = (P_B_given_A * P_A) / P_B
print(f"P(A|B) = {P_A_given_B:.4f}")  # Output: 0.1672

Interpretation: Even if you test positive, there is only about a 16.7% chance you actually have the disease.

Key Takeaways

Conditional probability finds the chance of A happening when we know B occurred;
Bayes' Theorem flips conditional probabilities to update beliefs when direct measurement is difficult;
Both are essential in data science, medical testing, and machine learning.

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Section 5. Chapter 4

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Conditional Probability

Conditional probability measures the chance of an event happening given another event has already occurred.

Formula:

P(A \mid B) = \frac{P(A \cap B)}{P(B)}


              12345
            
P_A_and_B = 0.1  # Probability late AND raining
P_B = 0.2        # Probability raining

P_A_given_B = P_A_and_B / P_B
print(f"P(A|B) = {P_A_given_B:.2f}")  # Output: 0.5

Interpretation: if it is raining, there's a 50% chance you will be late to work.

Bayes' Theorem

Bayes' Theorem helps us find $P(A|B)$ when it's hard to measure directly, by relating it to $P(B|A)$ which is often easier to estimate.

Formula:

P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)}

Where:

$P(A \mid B)$ - probability of A given B (our goal);
$P(B \mid A)$ - probability of B given A;
$P(A)$ - prior probability of A;
$P(B)$ - total probability of B.

Expanding $P(B)$

P(B) = P(B \mid A) P(A) + P(B \mid \neg A) P(\neg A)


              123456789101112
            
P_A = 0.01                # Disease prevalence
P_not_A = 1 - P_A
P_B_given_A = 0.99        # Sensitivity
P_B_given_not_A = 0.05    # False positive rate

# Total probability of testing positive
P_B = (P_B_given_A * P_A) + (P_B_given_not_A * P_not_A)
print(f"P(B) = {P_B:.4f}")  # Output: 0.0594

# Apply Bayes’ Theorem
P_A_given_B = (P_B_given_A * P_A) / P_B
print(f"P(A|B) = {P_A_given_B:.4f}")  # Output: 0.1672

Interpretation: Even if you test positive, there is only about a 16.7% chance you actually have the disease.

Key Takeaways

Conditional probability finds the chance of A happening when we know B occurred;
Bayes' Theorem flips conditional probabilities to update beliefs when direct measurement is difficult;
Both are essential in data science, medical testing, and machine learning.

Everything was clear?

Thanks for your feedback!

Section 5. Chapter 4