Orange or red-and-yellow?
This example shows how easily truthful eyewitness statements can be misinterpreted. It is inspired by a chapter in the book Der Schein der Weisen by Hans-Peter Beck-Bornholdt and Hans-Herrmann Dubben.
The teams of two football clubs, FC Schwalbe and FC Money-Cicker, meet in a match that is important to both clubs. FC Schwalbe wins. From the point of view of the FC Money-Cicker supporters’ club, however, this is due only to bad refereeing decisions and unfair play. Despite a strong police presence, riots break out during the night.
In a dark alley, Mr Dackelberger, who is walking his dog, sees someone being knocked to the ground and a figure in an orange jacket running away. Such jackets are worn by members of the hard core of the FC Money-Cicker supporters.
For the local press, the case seems clear: a supporter of FC Money-Cicker could not accept the defeat and beat a fan of the other club badly enough to send him to hospital. A reader points out, however, that in twilight an orange jacket can easily be confused with a red-and-yellow jacket of the kind worn by FC Schwalbe supporters.
According to his estimate, 100 FC Schwalbe hooligans and 20 FC Money-Cicker hooligans were in the town at the time of the offence. Without further information this would initially mean that 100 of 120 possible offenders — around 83 percent — belonged to FC Schwalbe.
Mr Fermat, a retired mathematics teacher, adds a decisive point: the eyewitness’s error rate. Suppose the witness identifies the jacket colour correctly in 80 percent of cases and is wrong in 20 percent.
| Jacket colour | Number | Witness believes it is red-yellow | Witness believes it is orange |
|---|---|---|---|
| orange | 20 | 20% of 20: 4 | 80% of 20: 16 |
| red-yellow | 100 | 80% of 100: 80 | 20% of 100: 20 |
Altogether, therefore, the witness believes he sees an orange jacket in 36 cases: correctly 16 times and incorrectly 20 times. Given that he reports “orange”, the probability of an error is therefore
20 / 36 ≈ 55.6%.
Although the witness identifies the colour correctly in 80 percent of all individual observations, a specific report of “orange” is more often wrong than right in this scenario. The reason is the very different base frequency of the two jacket colours. The example illustrates what is known as the base-rate fallacy.