#41 Bayesian Inference

Thinking, Fast and Slow (Daniel Kahneman, 2011)

Biases

A cab was involved in a hit-and-run accident at night.

Two cab companies, the Green and the Blue, operate in the city.

You are given the following data:

85% of the cabs in the city are Green and 15% are Blue.
A witness identified the cab as Blue.
The court tested the realiability of the witness under the circumstances that existed on the night of the accident and concluded that the witness correctly identified each one of the two colors 80% of the time and failed 20% of the time.

What is the probability that the cab involved in the accident was Blue rather than Green?

ACCIDENT
          │
          ├── GREEN cab (0.85)
          ├── Witness says BLUE  (0.20) → 0.17 → WRONG  → sum to BLUE total
          ├── Witness says GREEN (0.80) → 0.68 → CORRECT
          │
          ├── BLUE cab (0.15)
          ├── Witness says BLUE  (0.80) → 0.12 → CORRECT → sum to BLUE total
          ├── Witness says GREEN (0.20) → 0.03 → WRONG
            
          BLUE REPORT TOTALS
          │
          ├── From GREEN cab saying BLUE → 0.17
          └── From BLUE cab saying BLUE  → 0.12
          Total probability witness says BLUE → 0.29
              
          FINAL PROBABILITY CAB IS ACTUALLY BLUE (Bayes' Theorem)
          │
          ├── P(Blue|Witness says Blue) = P(Blue and witness says Blue) 0.12 / P(Witness says Blue) 0.29 ≈ 0.41

KARL BALD

She said blue. Case closed.

KUNST

My guy… they didn’t even ask that.

They’re asking about the probability it was blue, not your caveman instinct.

You skipped the base rates, the math, the logic, you just heard “blue” and went full autopilot.

KARL BALD

I go with my gut, man. Witness said blue, I trust her.

KUNST

Your gut has a 20% failure rate.

And that 80% accuracy? It's only working on the 15% chance the cab was Blue.

So don't think you're getting 80% certaintly overall, you're just getting 80% of a small slice.

KARL BALD

Damn. That’s cold.

KUNST

Not as cold as your logic, dude.

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#41 Bayesian Inference

Biases

LESSON 41