The Independence Fallacy
- Assuming Events are Independent When Such Might Not be the Case1
- Multiplying Non-Independent Probabilities2
Subfallacy: The Gambler's Fallacy
In December of 1996, Christopher Clark, the two-and-a-half week old son of Steve and Sally Clark, died unexpectedly. Initially, it was assumed that he died of natural causes. In January of 1998, the couple's second son, Harry, also died at about the same age under similar circumstances.
Despite the fact that there was no forensic evidence that either child had been murdered, the police arrested the Clarks on charges of killing their two sons. The charges against Steve Clark were soon dropped because he had alibis for both times of death, but in both cases Sally had been alone with the babies when they died. She was eventually tried on two counts of murder.
At the time, the incidence of sudden unexplained infant death―called "crib death" or "cot death"―was approximately one in 8,500 in middle-class families such as the Clarks. A pediatrician testified at the trial that, by multiplying this probability by itself, the probability of two such deaths was only one in about 73 million.
Despite the lack of evidence of murder, Sally Clark was convicted on November 9th, 1999, and sentenced to life imprisonment. Her conviction was overturned in 2003, so that she spent only three years in prison, but that was still too late for her. She drank herself to death on March 15, 2007 at the age of 42.4
In probability theory, two events are said to be "independent" when the occurrence of one does not affect the probability of the other. For instance, suppose that you draw a card from a well-shuffled full deck of playing cards. What is the probability that you draw an ace? Since there are four aces and fifty-two cards, the probability is four in fifty-two, or one in thirteen.
Suppose, now, that you return the ace to the deck, shuffle the deck, and draw a card again. What is the probability that you draw an ace? There are still four aces in a deck of fifty-two cards, so it's still one in thirteen. The fact that you drew an ace previously does not affect the probability that you draw one again, so these two events are independent.
In contrast, suppose that you don't return the ace to the deck after the first draw, but shuffle and draw again. What is the probability that you now draw an ace? Since the deck is reduced to fifty-one cards and three aces, the probability is three in fifty-one, which is only one in seventeen. So, the two events of drawing an ace from a full deck and then drawing one from a reduced deck are not independent. The occurrence of the first event affects the probability of the second.
An important law of probability theory is known as the Multiplication Rule: in English, it says that the probability that independent events occur is the product of the individual probabilities of each event. Symbolically: p(A ∩ B) = p(A) × p(B), where A and B are independent events. Here, "p(A)" is the probability that event A occurs and "A ∩ B" is the joint event of A and B both happening.5
Note that this rule will not give the correct probability if A and B are not independent events. The Independence Fallacy occurs when the Multiplication Rule is applied to events that are not independent, or are not known to be independent.
Analysis of the Example: In order to arrive at the probability of two cases of crib death in the Clark family, the pediatrician multiplied the probability of one such death―1 in 8,500―by itself to get one in 73 million6. It's at least doubtful that two such deaths were probabilistically independent since there is evidence that they tend to run in families, which could be due to shared genetics or environment7. As explained in the Exposition, above, it is incorrect to multiply non-independent probabilities, so that the pediatrician committed the Independence Fallacy by claiming that the probability of two crib deaths in the Clark family was one in 73 million.
- Stephen K. Campbell, Flaws and Fallacies in Statistical Thinking (1974), pp. 129-131 & 198.
- Leila Schneps & Coralie Colmez, Math on Trial: How Numbers Get Used and Abused in the Courtroom (2013), p. 1.
- This is my name for a common mistake that lacks a common name. The only alternatives I've found are the unwieldy aliases used by Campbell and Schneps & Colmez in the books cited in the two previous notes.
- This account of the Sally Clark case is based on the following sources: , BBC News, 11/7/2007
- Alicia Carriquiry, "Misuse of Statistics in the Courtroom: The Sally Clark Case", Center for Statistics and Applications in Forensic Evidence, 2/16/2018
- Helen Joyce, "Beyond reasonable doubt", Plus, 9/1/2002
- Leila Schneps & Coralie Colmez, Math on Trial: How Numbers Get Used and Abused in the Courtroom (2013), pp. 2-21
- The Multiplication Rule is a corollary of Axiom 4 in the system given in the entry for Probabilistic Fallacy; see Note 4 to that entry.
- More precisely, the probability is one in 72,250,000.
- See: "Royal Statistical Society concerned by issues raised in Sally Clark case", Royal Statistical Society, 10/23/2001.