Two cab companies operate in a given city, the Blue and the Green. 85% percent of the cabs in the city are Blue, and the remaining 15% are Green.

A cab was involved in a hit-and-run accident at night. A witness later identified the cab as a Green cab. The court tested the witness’ ability to distinguish between Blue and Green cabs under nighttime visibility conditions. It found that the witness was able to identify each color correctly about 80% of the time, but confused it with the other color about 20% of the time.

What do you think is the percent chance that the errant cab was indeed Green, as the witness claimed? (Please answer in the range 0 to 100).

Question

Two cab companies operate in a given city, the Blue and the Green. 85% percent of the cabs in the city are Blue, and the remaining 15% are Green.

A cab was involved in a hit-and-run accident at night. A witness later identified the cab as a Green cab. The court tested the witness’ ability to distinguish between Blue and Green cabs under nighttime visibility conditions. It found that the witness was able to identify each color correctly about 80% of the time, but confused it with the other color about 20% of the time.

What do you think is the percent chance that the errant cab was indeed Green, as the witness claimed? (Please answer in the range 0 to 100).

BlackTom AI · Accepted Answer

Apply Bayes’ theorem: the posterior probability that the cab was Green given the witness said “Green” equals P(report Green | Green)·P(Green) divided by the total probability the witness reports Green.

Relevant concepts:
- Prior probability (base rate): P(Green) = 0.15, P(Blue) = 0.85.
- Witness reliability: P(report correct) = 0.80, so P(report Green | Green) = 0.80 and P(report Green | Blue) = 0.20 (the witness confuses colors 20% of the time).
- Bayes’ theorem: P(G | report G) = P(report G | G)·P(G) / [P(report G | G)·P(G) + P(report G | B)·P(B)].

Step-by-step calculation with all intermediate steps:
1. Compute the numerator: P(report G | G)·P(G) = 0.80 × 0.15 = 0.120.
2. Compute the contribution from a Blue cab being reported Green: P(report G | B)·P(B) = 0.20 × 0.85 = 0.170.
3. Total probability the witness reports Green = 0.120 + 0.170 = 0.290.
4. Posterior probability = 0.120 / 0.290 ≈ 0.413793103.

Convert to percent: 0.413793103 × 100 ≈ 41.379%, which rounds to 41%.

Why the correct answer is 41%:
Although the witness is 80% accurate, the low base rate of Green cabs (15%) means many true Blues can be misidentified as Green. The Bayes calculation properly combines witness reliability with base rates, yielding about 41% probability the cab was actually Green.

Common incorrect intuitions:
- Answering 80% ignores base rates (base rate fallacy).
- Answering 15% ignores the witness report.
Both are incorrect because they use only one of the two necessary pieces of information.

Conclusion: The percent chance the errant cab was Green is approximately 41%, matching the provided answer.

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