ECON30019 Chap.8 Representativeness, Small Numbers and Base Rates
Representativeness, Small Numbers and Base Rates
The gambler's fallacy is the expectation that a run away from a system's usual behaviour will right itself soon, and underneath it is a simpler error: independent outcomes get treated as though one constrained the next. The chapter separates the two probabilities that get conflated, the small probability of a long run assessed in advance and the unchanged probability of the next trial given what has already landed.
It then covers the representativeness heuristic, which estimates probability by judging how representative an outcome is of the process, and shows why that ranks equally likely sequences differently. The law of large numbers is real but applies to long runs; people behave as though a law of small numbers held.
The second half turns to base rates, the three quantities a judgement should reflect, and the false-positive computation that dominates assessed work.
What this chapter covers
- 01
The gambler's fallacy, and the dependence assumption underneath it
- 02
Two probabilities that get conflated, one small and one unchanged
- 03
Negative and positive recency, and the case where continuation is rational
- 04
The representativeness heuristic as a substitution of resemblance for likelihood
- 05
Three sequences ranked by resemblance, and why they are equally likely
- 06
The law of large numbers, correctly stated
- 07
The belief in a law of small numbers, and the drawing-without-replacement model
- 08
Base rates, and the three factors a judgement should reflect
- 09
Base-rate neglect: general information ignored, specific information kept
- 10
The false-positive computation, and the counts that make it obvious
A rare fault and a good alarm
- 1Name the events and list the three given quantities: the base rate is 0.02, the probability of a flag when defective is 1, and the probability of a flag when sound is 0.10.
- 1Build the denominator as the total probability of a flag: one times 0.02 plus 0.10 times 0.98.
- 1Evaluate it: 0.02 plus 0.098, which is 0.118.
- 1Divide by the numerator term: 0.02 over 0.118, approximately 0.169.
- 1Convert to counts per hundred to explain: two defective components all flagged, against 9.8 flags from the ninety-eight sound ones.
- 1State the direction: the sound group is fifty times larger, so it contributes most of the flags even though each unit is unlikely to be flagged.
Key terms
- Gambler's fallacy
- The expectation that a run away from a system's usual behaviour will right itself soon, which treats independent trials as though one constrained the next.
- Representativeness heuristic
- A rule that judges how likely an outcome is by how closely it resembles the process behind it, so a more typical-looking outcome is rated more probable.
- Law of large numbers
- The result that as the number of trials increases, the observed proportion approaches the theoretical probability. It says nothing about short runs being corrected.
- Law of small numbers
- A name for the mistaken belief that small samples resemble the population closely, which is what generates the expectation of short-run correction.
- Base rate
- The fraction of a population that has the characteristic in question. It is the prior of Bayes' rule under a different name.
- Base-rate neglect
- Giving the base rate less weight than it deserves, so that general information about a population is ignored while specific information about a case is kept.
Representativeness, Small Numbers and Base Rates FAQ
If both expecting a reversal and expecting a streak are errors, what is the correct expectation?
With a process known to be independent, the next trial's probability is unchanged by everything that came before, so neither reversal nor continuation should be expected. The condition matters: with a device of unknown fairness, a long run is legitimate evidence that it is biased, so expecting continuation can be rational there. State that condition and you earn the mark.
Why does an accurate test produce such a low probability of the condition?
Because relatively few people or items actually have it. When the base rate is small, the very large group without the condition can generate more false positives in absolute numbers than the small group with it generates true ones, however good the test is. Intuitive answers in the demonstrations run as high as ninety-five per cent when the correct figure is a small fraction of that.
Is there a case worth memorising as a sanity check?
Yes. If the base rate is one in a hundred and the test is ninety-nine per cent accurate in both directions, the probability of the condition after a positive result is exactly one half: one true positive per hundred people, and one false positive from the ninety-nine who are clear. If a question's numbers are near that and your answer is near ninety-nine per cent, you have inverted the conditional.
Exam move
The single habit that carries this chapter is writing both conditionals out in words before substituting. When a question says a test or an inspector is highly accurate, that number is the probability of the signal given the state, and the question almost always wants the probability of the state given the signal.
Practise the computation until you can build the denominator without thinking, then practise converting the result into counts per hundred, because interpretation marks reward the count form. For the gambler's fallacy, rehearse the two probabilities side by side so that you can produce both in one line.
Working through Representativeness, Small Numbers and Base Rates in ECON30019? Sia is AskSia’s AI Economics tutor — ask any ECON30019 Representativeness, Small Numbers and Base Rates question and get a clear, step-by-step explanation grounded in how ECON30019 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
