Bayes Theorem
$$ P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} $$Where:
- P(A|B) : The probability of event \( A \) occurring given that \( B \) has occurred (this is called the “posterior”).
- P(B|A) : The probability of event \( B \) occurring given that \( A \) has occurred (called the “likelihood”).
- P(A) : The prior probability of event \( A \) occurring (before we know \( B \)).
- P(B) : The total probability of event \( B \) occurring (the normalizing constant, which accounts for all possible ways \( B \) could happen).
This formula allows you to update your belief about the probability of \( A \) after knowing that \( B \) has occurred. It’s the foundation of Bayesian reasoning.
Scenario:
- \( A \): Your friend responds quickly to a text message (within 5 minutes).
- \( B \): You sent a message in the morning.
You want to know the probability that your friend will respond quickly (\( A \)) given that you sent the message in the morning (\( B \)).
Given Data (Realistic Assumptions):
- \( P(A) \): The probability that your friend responds quickly to a message in general is 20%, so \( P(A) = 0.20 \).
- \( P(B|A) \): If your friend responds quickly, there’s a 70% chance that it happens in the morning. So, \( P(B|A) = 0.70 \).
- \( P(B) \): The general probability that you send a message in the morning is 50%, so \( P(B) = 0.50 \).
Question:
What is the probability that your friend will respond quickly, given that you sent the message in the morning? We are looking for \( P(A|B) \).
Apply Bayes’ Theorem:
$$ P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} $$Substituting the known values:
$$ P(A|B) = \frac{0.70 \cdot 0.20}{0.50} $$ $$ P(A|B) = \frac{0.14}{0.50} = 0.28 $$Conclusion:
The probability that your friend will respond quickly, given that you sent the message in the morning, is 28%.