Law of Total Probability Interview Questions
How to use the law of total probability in quant interviews by splitting cases, weighting conditional probabilities, and connecting to Bayes rule.
Candidates practicing hidden-state, case-split, and conditional probability prompts.
Split into a partition
The law of total probability works when cases form a complete, non-overlapping partition. Compute the probability of the target inside each case, then weight by the probability of each case.
Formula in words
The total probability of A is the probability of A through case 1 plus the probability of A through case 2, and so on. Each term is P(case) times P(A given case).
Concrete example
If one of two coins is chosen at random, and coin one has heads probability 0.5 while coin two has heads probability 0.8, the total probability of heads is 0.5(0.5) + 0.5(0.8) = 0.65.
Bayes connection
Bayes rule often uses total probability in the denominator. The denominator sums all ways the observed signal could have happened.
When to use it
Use total probability when a hidden state, first step, or scenario changes the probability of the target. The case split should simplify the target event.
Common mistakes
Candidates often use overlapping cases or forget to weight by case probabilities. Check that cases are complete and mutually exclusive.
Practice the pattern
Use the LeetQuidity curriculum and calibration to turn this topic into a focused practice plan.