Discussion 3: Common mistakes in conditional logic

Discussion 3: Common mistakes in conditional logic#

STATS 60 / STATS 160 / PSYCH 10

Conditional logic mistakes in the media#

The section’s top two examples of mistakes in conditional logic in the media:

Conditional logic game show!#

Rules of the game:

Form teams of 2, and choose an ordering of all of the teams.
Each of the following slides will have a statement which contains a mistake in conditional logic.
Before I reveal the slide, next two teams will face off.
I will read the example on the slide, then the teams race to
1. Model the scenario using the language of conditional probabilities: what are the events in question, what is the information/statistic phrased in the language of conditional probability, and what is the mistaken conclusion or implication?
2. Identify the logical mistake or fallacy.
The first team to finish raises their hands. If they are correct, they are awarded a bonus point. If they are wrong, the other team gets a bonus point.
We’ll then talk through the scenario as a class.

Scenario 1:#

Marijuana is a gateway drug: 9 out of 10 opiate addicts say Marijuana is the first drug they tried.

$M$ is the event of trying marijuana (ever, and in particular before any other drugs). $O$ is the event of being an opiate addict.

This is saying $\Pr[M \mid O] = 0.9$.

The statement that “Marijuana is a gateway drug” suggests that $\Pr[O \mid M]$ is similarly high.

But we cannot conclude that from this information!

This could be seen as either:

The prosecutor’s fallacy (confusing $\Pr[O \mid M]$ with $\Pr[M \mid O]$), or
The base rate fallacy (maybe $\Pr[O]$ is small to begin with, so $\Pr[O \mid M]$ could still be small).

Scenario 2:#

A Stanford professor interacts with their students, and concludes that young people these days are much more likely to be stressed out than the professor remembers from back in their day.

This is the fallacy of generalizing from a biased sample.

Let $U$ be the event that a person is a Stanford student. Let $S$ be the event that a person is stressed out.

The professor has observed $\Pr[S \mid U]$ to be high, and based on this assumes that $\Pr[S]$ is high (in the sample space of young people).

But this could be sample bias, as it could be that young people who are not Stanford students are not as stressed and $\Pr[S]$ is much smaller than $\Pr[S \mid U]$.

Scenario 3:#

A crime scene has footprints from size 14 men’s shoes; less than 5% of men have size 14 shoes.

The primary suspect wears size 14 shoes, but in the city where the crime took place there are at least 25,000 men with size 14 shoes, so there is only a 1/25000 chance that he committed the crime.

This is an example of the defense attorney’s fallacy.

Let $G$ be the event of having committed the crime, let $L$ be the event of having size 14 shoes.

The argument here is that, within the sample space of the city where the crime was committed,

$\Pr[G \mid L] = 1/25000$.

But presumably the primary suspect was not just chosen uniformly among the people who wear size 14 shoes; there is likely other evidence against him as well.

When conditioning on that additional evidence, the likelihood of guilt increases.

Scenario 4:#

Only 7% of people have type O-negative blood. At a crime scene of a murder, the blood of the victim (type A positive) was found to be mixed with type O-negative blood.

The chief suspect has type O-negative blood. So the chance that the suspect is the perpetrator is 93%.

Let $O$ be the event of the defendant having type O-negative blood. Let $G$ be the event of the defendant being guilty.

The argument is that if the man were innocent, the chance of having type O-negative blood is $\Pr[O \mid \overline{G}] = 0.07$.

The fallacy here is to assume that $\Pr[\overline{G} \mid O] = 0.07$, and therefore by the law of complements $\Pr[G \mid O] = .93$.

This is the prosecutor’s fallacy; we are actually interested in the chance the defendant is innocent, $$\Pr[\overline{G} \mid O].$$

Scenario 5:#

A recruiter is reviewing applications for a job, for which only 10% of the applicant pool is actually qualified.

The recruiter automatically marks any applicant with a GPA $\ge 3.7$ as “likely qualified”, reasoning that 3.7 is the GPA associated with an A- average or above, and an A- is a good grade.

This is an example of the base rate fallacy.

Let $Q$ be the event that the candidate is qualified, and let $A$ be the event of having a GPA at least 3.7.

The recruiter reasons that $\Pr[A \mid Q]$ is high. But the recruiter then goes on to assume that $\Pr[Q \mid A]$ must be similarly high.

This neglects the base rate for $Q$ and also the base rate for $A$!

\[\Pr[Q \mid A] = \Pr[A \mid Q] \cdot \frac{\Pr[Q]}{\Pr[A]}.\]

If almost everyone has a high GPA because of grade inflation, and only 10% of the applicant pool is qualified, then $\Pr[Q \mid A]$ would be small.