Bayes' theorem reverses the direction of conditional probability. Given partition $B_1$, $B_2$, ..., $B_n$ of the sample space:

P($B_i$|A) = P(A|$B_i$) * P($B_i$) / [sum over j: P(A|$B_j$) * P($B_j$)]

The denominator is the Total Probability: P(A) = sum P(A|$B_j$) * P($B_j$).

Step-by-step approach (BAIT):

1. Branches: Identify the partitioning events (e.g., which box, which machine)
2. Assign priors: Write P($B_i$) for each branch
3. Input likelihoods: Write P(A|$B_i$) — probability of observed event given each branch
4. Total and compute: Calculate denominator then apply formula

Common JEE pattern: "Given the item is defective, what is the probability it came from machine A?" This is a classic reverse conditional — use Bayes'.