Taxing question of the blue (or green) sideswiper

A taxi sideswiped a car on a misty night and drove off. Of the cabs in the town, 85 per cent are green and 15 per cent are blue. The witness says the cab was blue. The witness is tested under conditions like those on the night of the accident; she gets the cab's colour right 80 per cent of the time, regardless of whether it is green or blue. Given all this, the probability that the sideswiper is blue is: a) 80 per cent, b) between 79 per cent and 51 per cent, c) 50 per cent, or d) less than 50 per cent.

Ian Hacking begins his book on inductive logic with seven such intriguing examples. He discusses the nature of inductive logic, how to calculate probabilities, how to combine probabilities with utilities, conceptual approaches to probability (including frequency and actual/ rational belief views), and related philosophical problems.

Writing a comprehensive introductory textbook is difficult. It requires understanding a broad field and the interrelationship and relative importance of its parts. It demands clear explanations, illuminating examples, a fair treatment of controversial issues and stimulating exercises that show the relevance of the material.

Hacking's book excels in all these ways, but especially in the practical, concrete examples. It uses minimal mathematics and presumes no acquaintance with symbolic logic. It is well suited for graduate or advanced undergraduate courses in inductive logic or related areas (such as philosophy of science or methodology courses in particular empirical sciences).

The book gives a nice introduction to inductive logic. I was disappointed, however, that it did not mention maximin, analogical reasoning, or Nelson Goodman's grue-bleen problem; but these are minor points.

So what about the taxi? Surprisingly, the correct answer is d): it is more likely than not that the witness was wrong about the taxi being blue. Recall that 85 per cent of the town's taxis are green while 15 per cent are blue, and that the witness gets the colour right 80 per cent of the time. Suppose that the witness saw 100 taxis from the town. She would probably see about 85 green ones (and mistakenly report 20 per cent of these, or 17, as blue) and about 15 blue ones (and correctly report 80 per cent of these, or 12, as blue). Note the large number of "false positives"; of the 29 (17 plus 12) taxis that she would report as blue, only 41 per cent (12/29) would actually be blue. So it is less than 50 per cent likely that the sideswiper taxi was blue.

Harry Gensler is professor of philosophy, John Carroll University, Cleveland, US.