What is “conditional probability”?
Conditional probability is the probability that some A will occur given B that has already occurred.
What is its significance?
One of the applications of conditional probability is in law; it is part of a concept known as the Prosecutor’s Fallacy. This rests on the incorrect assumption that the probability that A will occur given B is the same as the probability that B will occur given A. The first, and most common, example is the Sally Clark case.
Sally Clark Case
Sally Clark had one son in 1996 who died fairly quickly after his birth. Again, she had a second son and he died quickly after his birth. She said that they both died of SIDS (Sudden Infant Death Syndrome), but she was still arrested for killing her two sons. During her trial, a statistician declared that the probability of a child dying of SIDS is 1 in 8500, and therefore the probability of two children dying of SIDS is 1 in 73 million ((1/8500)^2). This seems fairly persuasive, and in fact the jury thought so too and convicted her.
But, the statistician ignored conditional probability: it turns out, the probability that a second child will die of SIDS if the first one has already died of SIDS increases substantially. Additionally, the probability of a child dying of SIDS if the child is male is also much higher.
Secondly, the jury should have weighed the two possibilities: that of both children dying of SIDS and that of Sally Clark killing both her sons. It turns out that the probability that Sally Clark killed both her sons is much, much lower than the probability that they both died of SIDS. The jury, the lawyer, and the statistician did not consider these when arguing the case.
OJ Simpson Case
OJ Simpson was on trial, suspected of killing his ex-wife. There was lots of evidence against him, but the defense argued that because Simpson abused his wife, it was highly unlikely that he would kill her (the statistic was 1 in 2500 abusers kill their significant others). Therefore, to the jury, it seemed like OJ Simpson was likely to be innocent.
However, the pertinent statistic at this trial was not the one presented. As Leonard Mlodinow puts it, “The relevant number is not the probability that a man who batters his wife will go on to kill her (1 in 2,500) but rather the probability that a battered wife who was murdered was murdered by her abuser” (120). And, the relevant probability was about 9 in 10 abused women who were killed were killed by their abusers. Therefore, statistics was actually in favor of the prosecution, not the defense.
When thinking about probability, the lesson to take from these two cases is that it is very important to think about the relevant probability.
The next post in this series will discuss patterns and randomness!
Leonard Mlodinow, The Drunkard’s Walk