Direct answer

Core relationship: P(A given B) uses B as the new universe. Count the cases where both A and B occur, then divide by all B cases, not by the original total.

For conditional probability without formula confusion, begin with circle the given condition and finish with compare the reverse separately. The sequence keeps the starting facts, intermediate result, and final limit visible instead of relying on memory.

Method you can audit

Circle the given condition

Treat it as information already known. Capture the result before moving to restrict the denominator. A written output shows whether the next deduction follows from the evidence or quietly adds a new assumption.

Restrict the denominator

Only cases satisfying the condition remain in the reference group. Compare this output with the warning about using the full sample. If that failure still fits, revise the representation before using the result in another deduction.

Count the overlap

The numerator must satisfy both the target event and the condition. Use count the overlap as a checkpoint. Name the input, the change made, and the conclusion that becomes available so another reader can reproduce the move.

Compare the reverse separately

P(A given B) and P(B given A) usually have different denominators. Finish by stating what this step establishes and what it leaves undecided. That boundary prevents a valid local conclusion from being stretched beyond the question.

Check against using the full sample

Questions for restrict the denominator

  • Can another reader reproduce circle the given condition from the stated information?
  • Did you complete restrict the denominator without reversing or enlarging the claim?
  • Does the result survive the failure mode using the full sample?
  • Which part remains uncertain after compare the reverse separately?

This check is specific to the method above. A clear answer should expose the evidence chain and the point where a different fact would produce a different conclusion.

Worked interpretation

Circle the given condition in the case

Out of 100 orders, 20 are international and 8 of those are delayed. The probability of delay given international is 8 out of 20, or 40%, not 8 out of 100.

Why compare the reverse separately matters

The case starts with circle the given condition, then uses restrict the denominator to control the next inference. The conclusion is acceptable only after compare the reverse separately. Changing any fact that supports those moves can change the answer, while an unrelated detail should not.

Limits and traps

Using the full sample

That calculates a joint or unconditional probability instead. Return to circle the given condition and rebuild only from information the problem actually provides.

Reversing the condition

The same overlap does not make the two conditional rates equal. Compare the disputed move with count the overlap; the two should use the same definitions and direction.

Ignoring selection

The known condition may define a very different population. Run the practice task while deliberately watching for ignoring selection. Record the first point where the reasoning begins to drift.

Use it on a real problem

Build a two-by-two count table for delayed versus on-time and domestic versus international orders. Calculate both conditional directions and explain the denominator change.

Continue after compare the reverse separately

Use the probability reasoning guide for the broader method, then apply the idea in the probability calculator. Keep the written reasoning so the next attempt can improve a specific step rather than simply repeat the exercise.

Compare this method with Odd-One-Out Puzzles Need a Defensible Classification Rule, then continue to Base-Rate Neglect in Real Decisions for a different reasoning task.