Conditional Probability, Total Probability & Bayes' Theorem
“Prior to posterior updates, base-rate fallacy, partition of sample spaces, and conditional independence.”
1.Formal Probability Framework and Bayesian Inference
§ 4.1.Probability formalizes quantitative reasoning under uncertainty.
Sample Space and Events: A sample space S is the set of all possible outcomes. An event A is a subset of S.
Partitions: A family of events {A_1, A_2, ..., A_n} forms a partition of S if:
1. Pairwise disjoint: A_i intersect A_j = empty set for all i != j.
2. Exhaustive: Union_{i=1}^n A_i = S.
Bayes' Theorem Formulation:
Let {A_1, ..., A_n} be a partition of S with prior probabilities P(A_i) > 0. For an observed evidence event B with P(B) > 0:
P(A_k | B) = [ P(B | A_k) P(A_k) ] / [ Sum_{i=1}^n P(B | A_i) P(A_i) ]
- P(A_k) is the prior probability of hypothesis A_k.
- P(B | A_k) is the likelihood of evidence B given A_k.
- The denominator is the marginal evidence P(B).
- P(A_k | B) is the posterior probability after observing B.
If algebraic fraction manipulation gets confusing under exam pressure, assume a population of 100,000 individuals and build a two-tier branch tree with exact integer counts. The ratio of (target branch) / (all matching branches) gives the exact posterior probability without decimal errors.
2. Worked Problem: Three-Machine Defect Rate Bayes Analysis
Carefully map which machine produced the defective item vs which machine is being queried in the numerator.
Problem [Original Practice Problem · Modeled on GATE Pattern]:
In a factory, machines M1, M2, and M3 manufacture 30%, 25%, and 45% of total computer memory chips, respectively.
The defective output rates are 2% for M1, 3% for M2, and 1% for M3.
A chip is picked at random and found to be defective. What is the probability that it was manufactured by machine M2?
Step 1: Define events and identify prior probabilities:
P(M1) = 0.30, P(M2) = 0.25, P(M3) = 0.45
Verify partition: 0.30 + 0.25 + 0.45 = 1.00.
Step 2: Identify conditional defect likelihoods P(D | Mi):
P(D | M1) = 0.02, P(D | M2) = 0.03, P(D | M3) = 0.01.
Step 3: Compute total probability of defect P(D):
P(D) = P(D|M1)P(M1) + P(D|M2)P(M2) + P(D|M3)*P(M3)
P(D) = (0.02 0.30) + (0.03 0.25) + (0.01 * 0.45)
P(D) = 0.0060 + 0.0075 + 0.0045 = 0.0180 (1.8% defect rate overall).
Step 4: Apply Bayes' Formula for P(M2 | D):
P(M2 | D) = [ P(D | M2) * P(M2) ] / P(D)
P(M2 | D) = 0.0075 / 0.0180 = 75 / 180 = 5 / 12 ≈ 0.4167 (41.67%).
Primary Academic References & Exam Sources
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Efficient GATE maths coverage with high-yield topic prioritization.
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