Use the law of total probability by conditioning on the occupancy state (both occupied, exactly one occupied, both empty). For each state, compute the probability that a randomly chosen room is occupied, then take the weighted average using the given prior probabilities.
Pro tip: After computing the answer, briefly mention that this is a simple application of conditional expectation and that the result aligns with intuition: on average, half the rooms are occupied, so the probability is 1/2.
Let O be the event that the checked room is occupied. Define the occupancy states: A = both occupied, B = exactly one occupied, C = both empty.
Given P(A) = 1/3, P(B) = 1/3, P(C) = 1/3.
For each state, find P(O|state): P(O|A) = 1 (both occupied), P(O|B) = 1/2 (one occupied out of two), P(O|C) = 0 (both empty).
Compute P(O) = P(O|A)P(A) + P(O|B)P(B) + P(O|C)P(C) = 1*(1/3) + (1/2)*(1/3) + 0*(1/3) = 1/3 + 1/6 = 1/2.
The probability is 1/2, meaning there is a 50% chance the checked room is occupied.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
First, clarify the problem setup and define the events and prior probabilities. Then apply Bayes' theorem step-by-step, explicitly stating the formula and computing each term. Finally, interpret the result in the context of the problem and discuss any assumptions or implications.
Pro tip: Always state your assumptions clearly, especially about independence and prior probabilities, as this shows rigor and prevents misunderstandings. Also, relate the calculation to a real-world scenario to demonstrate practical insight.
Restate the question in your own words and specify the events: e.g., let A be the event that the other room is occupied, and B be the event that the chosen room is occupied. State any assumptions about independence or prior probabilities.
Assign prior probabilities P(A) and P(B) based on the given context or reasonable assumptions. If not given, discuss how you would estimate them (e.g., from historical data or symmetry).
Write Bayes' formula: P(A|B) = P(B|A) * P(A) / P(B). Compute the numerator and denominator, using the law of total probability for P(B) if needed.
Plug in the numbers to get the posterior probability. Interpret the result in plain language, explaining what it means for the likelihood that the other room is occupied given that the chosen room is occupied.
Mention how the result changes with different priors or assumptions, and relate it to similar problems in product analytics or root cause analysis.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.