Step 1: Identify the partition $\frac{causes}{sources}$ Common setups: multiple bags, machines, factories, or coin types. These are the $B_i$ events.

Step 2: Assign prior probabilities P($B_i$) Often given directly (e.g., "bag chosen with probability 1/3") or derived from proportions.

Step 3: Determine likelihoods P(A|$B_i$) The probability of the observed event given each cause. E.g., P(red|Bag 1) = 3/7.

Step 4: Compute total probability P(A) P(A) = sum P(A|$B_i$)*P($B_i$). This is the denominator.

Step 5: Apply Bayes' formula P($B_i$|A) = P(A|$B_i$)*P$\frac{B_i}{P}$(A)

Example template: Three machines produce 20%, 30%, 50% of items with defect rates 5%, 3%, 2%.

• P($B_1$)=0.2, P($B_2$)=0.3, P($B_3$)=0.5
• P(D|$B_1$)=0.05, P(D|$B_2$)=0.03, P(D|$B_3$)=0.02
• P(D) = 0.01+0.009+0.01 = 0.029
• P($B_1$|D) = 0.01/0.029 = 10/29

Common traps:

1. Confusing P(A|B) with P(B|A) — direction matters
2. Forgetting to use total probability for the denominator
3. Not partitioning the sample space correctly