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 hitand-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 hitand-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