17.2: Definition and Notation

The expression \(\text{Pr}[X \mid Y]\) denotes the probability of event \(X\), given that event \(Y\) happens. In the example above, event \(X\) is the event of winning on a switch, and event \(Y\) is the event that a goat is behind door B and the contestant chose door A. We calculated \(\text{Pr}[X \mid Y]\) using a formula which serves as the definition of conditional probability:

Definition \(\PageIndex{1}\)

Let \(X\) and \(Y\) be events where \(Y\) has nonzero probability. Then

\[\text{Pr}[X \mid Y] ::= \dfrac{\text{Pr}[X \cap Y]}{\text{Pr}[Y]}.\]

The conditional probability \(\text{Pr}[X \mid Y]\) is undefined when the probability of event \(Y\) is zero. To avoid cluttering up statements with uninteresting hypotheses that conditioning events like \(Y\) have nonzero probability, we will make an implicit assumption from now on that all such events have nonzero probability.

Pure probability is often counterintuitive, but conditional probability can be even worse. Conditioning can subtly alter probabilities and produce unexpected results in randomized algorithms and computer systems as well as in betting games. But Definition 17.2.1 is very simple and causes no trouble—provided it is properly applied.

What went wrong

So if everything in the opening Section 17.1 is mathematically sound, why does it seem to contradict the results that we established in Chapter 16? The problem is a common one: we chose the wrong condition. In our initial description of the scenario, we learned the location of the goat when Carol opened door B. But when we defined our condition as “the contestant opens A and the goat is behind B,” we included the outcome \((A,A,C)\) in which Carol opens door C! The correct conditional probability should have been “what are the odds of winning by switching given the contestant chooses door A and Carol opens door B.” By choosing a condition that did not reflect everything known. we inadvertently included an extraneous outcome in our calculation. With the correct conditioning, we still win by switching 1/9 of the time, but the smaller set of known outcomes has smaller total probability:

\[\nonumber \text{Pr}[\{(A, A, B), (C, A, B)\}] = \frac{1}{18} + \frac{1}{9} = \frac{3}{18}.\]

The conditional probability would then be:

\[\nonumber \text{Pr}[[\text{[win by switching}] \mid [\text{[pick A AND Carol opens B}]] = \text{Pr}[(C, A, B) \mid \{(C, A, B), (A, A, B)\}] + \dfrac{\text{Pr}[(C, A, B)]}{\text{Pr}[\{(C, A, B), (A, A, B)\}]} = \dfrac{1/9}{1/9 + 1/18} = \dfrac{1}{2}.\]

which is exactly what we already deduced from the tree diagram 16.2 in the previous chapter.

The O. J. Simpson Trial

In an opinion article in the New York Times, Steven Strogatz points to the O. J. Simpson trial as an example of poor choice of conditions. O. J. Simpson was a retired football player who was accused, and later acquitted, of the murder of his wife, Nicole Brown Simpson. The trial was widely publicized and called the “trial of the century.” Racial tensions, allegations of police misconduct, and new-at-the-time DNA evidence captured the public’s attention. But Strogatz, citing mathematician and author I.J. Good, focuses on a less well-known aspect of the case: whether O. J.’s history of abuse towards his wife was admissible into evidence.

The prosecution argued that abuse is often a precursor to murder, pointing to statistics indicating that an abuser was as much as ten times more likely to commit murder than was a random indidual. The defense, however, countered with statistics indicating that the odds of an abusive husband murdering his wife were “infinitesimal,” roughly 1 in 2500. Based on those numbers, the actual relevance of a history of abuse to a murder case would appear limited at best. According to the defense, introducing that history would make the jury hate Simpson but would lack any probitive value. Its discussion should be barred as prejudicial.

In other words, both the defense and the prosecution were arguing conditional probability, specifically the likelihood that a woman will be murdered by her husband, given that her husband abuses her. But both defense and prosecution omitted a vital piece of data from their calculations: Nicole Brown Simpson was murdered. Strogatz points out that based on the defense’s numbers and the crime statistics of the time, the probability that a woman was murdered by her abuser, given that she was abused and murdered, is around 80%.

Strogatz’s article goes into more detail about the calculations behind that 80% figure. But the real point we wanted to make is that conditional probability is used and misused all the time, and even experts under public scrutiny make mistakes.