Consider this scenario: A taxi was involved in a hit-and-run accident at night. The city in which the accident took place has green and blue taxis. Green taxis comprise 90% of the taxis in the city, with blue taxis comprising 10%. A witness to the accident said that the taxi was blue. The reliability of the witness was tested under similar conditions to the night of the accident. For each color, the witness identified the correct color 80% of the time and identified the wrong color 20% of the time.
You will determine the updated probability, given that the witness identified the taxi as blue, that the taxi involved in the accident actually was blue. You will do this by completing a table of hypothetical counts, representing a total of 100 taxis.
a) Fill in the following table of hypothetical counts, in the order indicated:
Witness said that taxi was green | Witness said that taxi was blue | Total Taxi was actually green (iii) (iv) (i)
Taxi was actually blue (v) (vi) (ii) Total (vii) (viii) 100 Show your work for each of the eight steps, and also submit the completed table of counts. b) Use the completed table to determine the updated probability, given that the witness identified the taxi as blue, that the taxi involved in the accident actually was blue. Report your answer as a fraction and also as a decimal with three decimal places of accuracy, c) Given that the witness identified the taxi as blue, is it more likely that the taxi was actually blue or green? Explain why this makes sense, based on the most relevant information from the paragraph that described the scenario. (vi) d) Calculate the ratio of the updated probability that the taxi was blue, given that the witness identified the taxi as blue, to the original (prior) probability that the taxi was blue, before hearing from the witness. Also write a sentence to interpret this ratio in this context. For parts e) and f), suppose that the witness had identified the taxi as green. e) Use the completed table to determine the updated probability, given that the witness identified the taxi as green, that the taxi involved in the accident actually was green. Report your answer as a fraction and also as a decimal with three decimal places of accuracy. f) Comment on how this probability (that the taxi was actually green) has changed from its original (before hearing from the witness) probability to its updated after hearing that the witness identified the taxi as green) probability. For parts g) and h), suppose that instead of the 90/10 breakdown of green/blue taxis in the city, the breakdown of the two colors had actually been 50/50. Continue to assume that the witness reliability is 80% for each color. g) Determine the updated probability that the taxi was actually blue given that the witness identified it as blue. Also determine the updated probability that the taxi was actually green, given that the witness identified it as green. Also submit the completed table behind these calculations. h) Write a sentence or two to describe how the results that start with a 50/50 color breakdown of taxis in the city differ from those with a 90/10 color breakdown.