The Case of the Blue Cab and the Black Cab Companies * Apologies to the young *Adapted from:

The “Facts” Two cab companies Black cab company has 85 cabs Blue cab company has 15 cabs

What is the probability I see a blue cab?* A slight detour into probability theory

What is the probability I see a blue cab? P(I see a blue cab) = Total # blue cabs ___________________________ Total # cabs

What is the probability I see a blue cab? P(I see a blue cab) = Total # blue cabs ___________________________ Total # cabs 15/100=0.15

The “Facts” Two cab companies Black cab company has 85 cabs Blue cab company has 15 cabs The eye witness saw a blue cab in a hit- and-run accident at night

Can we trust the eye witness?

At the request of the defense attorney The eye witness under goes a ‘vision test’ under lighting conditions similar to those the night in question

The Vision Test Repeatedly presented with a blue taxi and a black taxi, in ‘random’ order.

The Vision Test Repeatedly presented with a blue taxi and a black taxi, in ‘random’ order. The eye witness shows he can successfully identify the color of the taxi for times out of five (80% of the time).

Would you find the blue taxi company guilty of hit and run?

How can we use the new information about the accuracy of the eye witness? Bayesian probability theory If the eye witness reports seeing a blue taxi, how likely is it that he has the color correct?

How can we use the new information about the accuracy of the eye witness? (cont) Eye witness is correct 80% of the time (4 out of 5) Eye witness is incorrect 20% of the time (1 out of 5)

How many blue taxis would he identify as correct?

How many blue taxis would he identify as correct/incorrect? (.8) * 15 = 12 (correct, i.e. blue) (.2) * 15 = 3 (incorrect, i.e. black)

How many black taxis would he identify as incorrect? (.2) * 85 = 17 (incorrect, i.e. blue)

Summary Misidentified the color of 20 taxis Identified 29 taxis as blue, even though there are only 15 blue taxis Probability that the eyewitness claimed the taxi to be blue actually was blue given the witness’s id ability is

Summary Misidentified the color of 20 taxis Identified 29 taxis as blue, even though there are only 15 blue taxis Probability that the eyewitness claimed the taxi to be blue actually was blue given the witness’s id ability is 12/29, i.e Incorrect nearly 3 out of five times

Bayesian probability takes into account the real distribution of the taxis in the town. Ability of the eye witness to identify the blue taxi color correctly Ability to identify the color of the blue taxis among all the taxis in town.

Would you find the blue taxi cab company responsible for the hit- and-run?

Discrete Bayes Formula* P(A|B) = P(B|A)P(A) P(B) conditional probability

Conditional Probability P(A|B) B

Conditional Probability P(A|B) B P(A and B)

Conditional Probability P(A|B) = P(A and B) P(B) B P(A and B)

For our taxi case P(taxi is blue| witness said blue)= P(witness said it was blue|taxi is blue)* P(taxi was blue) P(witness said it was blue)

For our taxi case P(taxi is blue| witness said blue)= P(witness said it was blue|taxi is blue)* P(taxi was blue) P(witness said it was blue) = (0.8)(15/100) = 0.41 (29/100)

Homework Work out the Monty Hall problem that Dale described yesterday