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Conditional Probability & Bayes’ Theorem

Updated on June 25, 2025
Business Statistics

Why it is important? 

Because not everything in life is independent. When one event affects another, we step into the world of conditional probability  a cornerstone of reasoning under uncertainty.

Unit 5: Business Statistics and Research Methods

Conditional Probability

Conditional probability is the probability of an event occurring, given that another event has already occurred. It's like asking, "What’s the chance of rain given that it’s cloudy?"

The formula is:

P(A|B) = P(A ∩ B) / P(B), where:

  • P(A|B) is the probability of A occurring given B has occurred.
  • P(A ∩ B) is the probability of both A and B occurring.
  • P(B) is the probability of B.
Example: Suppose 60% of students in a college are girls, and 20% of the girls are studying commerce. What is the probability that a randomly selected student is studying commerce given that the student is a girl?

Here, A = student studying commerce, B = student is a girl
P(A ∩ B) = 0.20 × 0.60 = 0.12
P(B) = 0.60
P(A|B) = 0.12 / 0.60 = 0.20

Independent vs Dependent Events

  • Independent Events: The occurrence of one does not affect the probability of the other. For example, tossing a coin twice.
  • Dependent Events: One event influences the probability of the next. For example, drawing cards from a deck without replacement.

Derivation of Bayes’ Theorem

Bayes’ Theorem is a mathematical formula used to update probabilities based on new evidence. It reverses the conditional probability.

Formula:

P(A|B) = [P(B|A) × P(A)] / P(B)

Let’s break it down:

  • P(A) is the prior probability of A
  • P(B|A) is the likelihood of B given A
  • P(B) is the total probability of B
  • P(A|B) is the updated (posterior) probability

Tree Diagram for Conditional Probability

Tree diagrams help visualize outcomes. Here's a basic illustration:

  • Start with two branches for event A and A’
  • From each, split into B and B’
  • Multiply probabilities along paths to find joint probabilities

Step-by-Step Example of Bayes’ Theorem

Problem: A disease affects 1% of a population. A test detects the disease 95% of the time if the person has it, and falsely signals 5% of the time if the person doesn’t. What is the probability that a person who tests positive actually has the disease?

Let D = has disease, T = tests positive
P(D) = 0.01, P(¬D) = 0.99
P(T|D) = 0.95, P(T|¬D) = 0.05

First, calculate P(T):
P(T) = P(T|D)×P(D) + P(T|¬D)×P(¬D)
P(T) = (0.95×0.01) + (0.05×0.99) = 0.0095 + 0.0495 = 0.059

Now, apply Bayes’ Theorem:
P(D|T) = (0.95×0.01) / 0.059 ≈ 0.161

So, even after testing positive, the chance of actually having the disease is only ~16.1%!

Real-World Applications

  • Marketing: What's the likelihood a customer will purchase given they clicked an ad?
  • Medical Testing: As seen above, to avoid false positives or negatives.
  • Risk Assessment: Calculating insurance probabilities based on prior events.

Conditional Probability vs Baye's Theorem

Aspect Conditional Probability Bayes’ Theorem
Direction Forward-looking (P(A|B)) Backward-looking (P(B|A))
Use Case Given event B, find A’s probability Update prior beliefs with evidence
Visual Tool Tree diagram, two-way tables Tree diagram, probability model

Probability isn’t just about flipping coins. When the world gets messy, and one thing depends on another, we need conditional probability and Bayes' Theorem to sort it out.

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