# FMM15

### From charlesreid1

# Friday Morning Math Problem

## Which Color Cab

The psychologists Daniel Kahneman and Amos Tversky used the following example to demonstrate the common failure to evaluate objective probabilities.

A cab is involved in a hit-and-run accident at night. Two cab companies, Blue and Green, operate in a city. You are given the following data:

A) 85% of cabs are green, 15% are blue

B) A witness identified the cab as blue. The reliability of the witness under the given conditions is that they correctly identify the color of the cab 80% of the time.

What is the probability that the cab the witness saw is actually blue?

Solution |
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We start by calculating two quantities:
1) The probability that the cab's apparent color was correctly identified 2) The probability that the cab's true color was correctly identified Case 1 is the superset, Case 2 is a subset. By taking the ratio of these two, we get the percent of time that the witness, identifying the cab color as blue, is correct because the cab was truly blue. Quantity 1: P(cab's apparent color correctly IDed) = P(cab actually blue) P(blue cab IDed as blue) + P(cab actually green) P(green cab IDed as blue) = (.15)(.80) + (.85)(.20) = (.12) + (.17) = .29 Quantity 2: P(cab's true color correctly IDed) = P(cab actually blue) P(blue cab IDed as blue) = (.15)(.80) = .12 Ratio of these two quantities is the percent of time the ID of a blue cab is because cab is actually blue: Ratio = 41.4% |

## Flags

Friday Morning Math ProblemsSpider Socks and Shoes FMM1 Sums of Powers of 2 FMM2 Fifty Coins FMM3 The Zeta Monogram FMM4 The Cthulhus Monogram FMM4B Multiplication Logic FMM5 The Termite and the Cube FMM6 Sharing Dump Trucks FMM7 The Flippant Juror FMM8 Bus Routes FMM9 A Robust Bus System FMM9B Square-Free Sequence FMM10 Inferring Rule from Sequence FMM11 Checkerboard Color Schemes FMM12 One-Handed Chords FMM13 First Ace FMM14 Which Color Cab FMM15 Petersburg Paradox Revisited FMM16 A Binomial Challenge FMM17 A Radical Sum FMM18 Memorable Phone Numbers FMM19 Arrange in Order FMM20 A Pair of Dice Games: FMM21
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