Blue-green cab problem

The Blue-green cab problem is a canonical demonstration of the psychological difficulty of Bayesian inference. It places a statistical fact about a population beside specific but imperfect evidence about an individual case, then shows how intuitive judgment can neglect the first and overvalue the second. The problem becomes especially revealing when an equivalent version turns the population statistic into a causal explanation: the mathematics stays the same, but the human response changes sharply.

The Basic Accident Scenario

A cab was involved in a hit-and-run accident at night in a city served by Green and Blue cab companies. Green cabs made up 85 percent of all cabs, while Blue cabs made up only 15 percent. A witness identified the vehicle as Blue. Testing conducted under the conditions of the accident found that the witness correctly identified each color 80 percent of the time and made an error 20 percent of the time.

The question asks for the probability that the cab involved was Blue rather than Green. The witness's report is relevant, but it cannot be treated as certain. The testimony has diagnostic value because a correct identification is more likely when the cab is Blue than when it is Green, yet the witness is also wrong one time in five. The population distribution therefore remains part of the calculation rather than disappearing once a case-specific observation is supplied.

Bayesian Combination of Evidence

Bayesian reasoning requires the base rate and the evidence to be combined. Before hearing the witness, the probability that the cab was Blue is 15 percent, simply because Blue cabs constitute 15 percent of the city's fleet. The witness then changes that prior belief, but does not erase it. The correct posterior estimate is about 41 percent, not the much higher probability that an immediate reading of the testimony may suggest.

The problem belongs to the wider logic of Bayes's rule, which combines prior beliefs with the diagnosticity of evidence. A description or observation may favor one hypothesis over another without making that hypothesis probable in isolation. The relevant question is not merely whether the witness's report fits a Blue cab; it is also how often such a report would arise when the cab was actually Green, and how common each color was before the report arrived.

Statistical and Causal Base Rates

The problem has a second version in which the two companies operate the same number of cabs, but Green cabs are involved in 85 percent of accidents and Blue cabs in only 15 percent. The witness's reliability remains unchanged. These two versions are mathematically indistinguishable: in both, the base rate for the cab involved in the accident is 15 percent, and the witness is correct 80 percent of the time. Yet people respond differently.

In the first version, the base rate is a statistical fact about the city's cab population. It says how many Green and Blue cabs exist, but it does not appear to explain why this particular driver committed a hit-and-run. A mind seeking a causal story finds little to use in the bare population statistic, so the witness's vivid, case-specific identification tends to dominate.

The second version gives the same numerical information a causal shape. If Green cabs are involved in more than five times as many accidents as Blue cabs, readers can imagine Green drivers as unusually reckless and apply that stereotype to the unknown driver in the case. The accident itself now supports a story pointing toward Green, while the witness's testimony supports a story pointing toward Blue. Because the two causal narratives conflict, their implications approximately cancel; judgments move toward equality, close to the Bayesian estimate of 41 percent for Blue.

Psychological Significance

The Blue-green cab problem distinguishes two kinds of base rates. Statistical base rates describe a population to which a case belongs, whereas causal base rates alter the account of how the individual case came about. Statistical base rates are generally underweighted, and may be neglected altogether when specific information about the case is available. Causal base rates are more readily treated as information about the individual and are therefore combined with other details of the case.

The contrast exposes a broader tension between probability and narrative. The witness's statement is concrete and easy to imagine; the fleet composition is abstract and impersonal. When the abstract statistic is translated into a causal account of reckless Green drivers, it becomes psychologically vivid and receives more weight. The resulting judgment is not a change in the evidence but a change in the way the evidence is represented.

The problem therefore serves as a compact discipline for intuition. A plausible story about the case should not be allowed to replace the population information that constrains the inference. Bayesian reasoning asks the judge to preserve both: the prior distribution of possible causes and the imperfect reliability of the new evidence. Its lasting lesson is that evidence feels more persuasive when it explains an individual event, but statistical reasoning must still account for what was probable before the story began.