S5: Misuse of Models and Multiplication Laws
Taken from Freedman Pisani and Purves -- Statistics
People v. Collins is a law case in which there was a major statistical issue. A black man and a white woman were charged with robbery. The facts were de-
scribed by the court as follows.
On June 18, 1964, about 11:30 A.M. Mrs. Juanita Brooks, who had been shopping, was walking home along an alley in the San Pedro area of the City of Los Angeles. She was pulling behind her a wicker basket carryall containing groceries and had her purse on top of the packages. She was using a cane. As she stooped down to pick up an empty carton, she was suddenly pushed to the ground by a person whom she neither saw nor heard approach. She was stunned by the fall and felt some pain. She managed to look up and saw a young woman running from the scene. According to Mrs. Brooks the latter appeared to weigh about 145 pounds, was wearing "something dark," and had hair "between a dark blond and a light blond," but lighter than the color of defendant Janet Collins' hair as it appeared at trial. Immediately after the incident, Mrs. Brooks discovered that her purse, containing between $35 and $40, was missing.
About the same time as the robbery, John Bass, who lived on the street at the end of the alley, was in front of his house watering his lawn. His attention was attracted by "a lot of crying and screaming" coming from the alley. As he looked in that direction, he saw a woman run out of the alley and enter a yellow automobile parked across the street from him. He was unable to give the make of the car. The car started off immediately and pulled wide around another parked vehicle so that in the narrow street it passed within six feet of Bass. The latter then saw that it was being driven by a male Negro, wearing a mustache and beard. At the trial Bass identified defendant as the driver of the yellow automobile. However, an attempt was made to impeach his identification by his admission that at the preliminary hearing he testified to an uncertain identification at the police lineup shortly after the attack on Mrs. Brooks, when defendant was beardless. In his testimony Bass described the woman who ran from the alley as a Caucasian, slightly over five feet tall, of ordinary build, with her hair in a dark blond ponytail, and wearing dark clothing. He further testified that her ponytail was "just like" one which Janet had in a police photograph taken on June 22, 1964.
The prosecutor then had a mathematics instructor at a local state college explain the multiplication rule, without paying much attention to independence, or the distinction between conditional and unconditional probabilities. After this testimony, the prosecution assumed the following chances:
Yellow automobile 1/10 Woman with blond hair 1/3
Man with mustache 1/4 Black man with beard 1/10
Woman with ponytail 1/10 Interracial couple in car 1/1,000
When multiplied together, these come to 1 in 12,000,000. According to the prosecution, this procedure gave the chance "that any [other] couple possessed the distinctive characteristics of the defendants." If no other couple possessed these characteristics, the defendants were guilty. The jury convicted. On appeal, the Supreme Court of California reversed the verdict.
QUESTION: EXPLAIN WHY Mathematics Instructors reasoning was wrong? check your ANSWER