Last week, I posed two questions related to the Business Insider headline about swearing and gossipping in the workplace (link). Today, I provide my answer to the first question. The answer to the second question is here.

**Question #1: Are the following headlines identical (i.e. interchangeable)?**

Answer: They are not identical.

More precisely, the two statements can be identical under limited conditions but they are **usually** not interchangeable. As Scott quipped on Twitter, it depends.

A reason why we may think those headlines say the same thing is that people tend to draw the same causal conclusion from each headline: if team members swear and gossip more, their chance of success increases.

We can unpack the statements. The key difference between them is the order of presentation. The first statement starts with the base of teams that are rated as the most successful by some metric of success, and within this base, there is a high incidence of swearing and gossiping. The second statement starts with the base of teams that engage in swearing and gossiping, and within this base, there is a high incidence of success.

For those familiar with conditional probabilities, the first statement concerns the probability of swearing and gossiping given that a team is "most successful" while the second statement is about the probability of success given a team engages in swearing and gossipping. In general, P(X/Y) is not the same as P(Y/X).

According to the famous Bayes' Theorem, the ratio of those conditional probabilities is equal to the ratio of the prior probabilities so the two statements are identical if and only if the rate of success among all teams is the same as the rate of swearing and gossipping. P(swear and gossip) = P(success) is an unlikely event in real life.

***

Let's bring back the helpful diagram from the answer to Question #2, that lays out the four possible characterizations of any team.

Consider an extreme case in which all teams are either in A or D. Every team that swears and gossips is successful while every team that doesn't fails. So B and C both are empty.

In this case, A+B (teams that swear) and A+C (teams that are successful) are identical. The prior probabilities, and also the conditional probabilities are the same. We conclude that the most successful teams swear and gossip, and also that the teams that swear and gossip are successful.

In the above diagram, the conditional probability of swear and gossip given success is A /(A+C) and that of success given swear and gossip is A/(A+B). For those to be identical, B and C must contain the same number of teams. In the extreme case, B = C = empty. If we put x teams in B, then we must put x teams in C, and then we have to subtract x teams from A and from D to make things even.

There are a small number of scenarios in which this can happen, and a much larger number of scenarios in which the two probabilities aren't the same.

***

Consider the two statements in this format:

Formula (I) comes from the original headline while formula (II) corresponds to the other headline.

Formula (II) predicts success (... leads to success) while formula (I) describes success (a successful team does ...). Formula (II) does not imply formula (I).

In the extreme case, every team swears and gossips, and so it follows that the most successful teams swear and gossip. Since everyone does it, swearing and gossiping will not predict success. Formula (II) does not imply formula (I).

In order to learn Formula (II), we must add other factors that affect team success.

(In the above, I excluded the possibility that success may cause swearing. Allowing this relationship complicates the causal interpretation even more.)

***

The confusion between descriptive and predictive (or causal) statements is endemic. For example, in coronavirus news, we keep hearing statements such as "Most of the hospitalized people are unvaccinated." This is very different from the related statement of "Most of the unvaccinated people are hospitalized."

If we think of vaccination status as a cause of an effect, and hospitalization as an outcome, then "Most of the hospitalized people are unvaccinated" is a description of people with a given outcome. It is not a predictive statement supporting a claim that vaccination status predicts some outcome. In addition, the first statement does not imply the second statement - that "most unvaccinated people are hospitalized."

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