The Case of the Blue Cab and the Black Cab

Companies*

Apologies to the young

*Adapted from: http://www.abelard.org/briefings/bayes.htm

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. 0.41

• 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