My wife had a small competition at her work. Fifteen of her 19 colleagues participated, divided into three teams of five. When she came home, she smiled and asked me, “Guess who was on the winning team?”

Given that my wife asked me the question, it’s highly likely she was on the winning team. But I like to practise thinking in terms of the base rate and not just the conditional probability1¹ See Meehl and the research inspired by his for why this is a good thing to do. so I answered “Peter”. Why Peter? I happen to know there are four (!) people named Peter at her job.

But what was the probability of Peter being on the winning team? At first, this sounded like a tricky question to answer, for at least two reasons:

1. Not all of her colleagues participated. Maybe all Peters did, but maybe none of the Peters did, or, most likely, something in between.
2. It matters how the Peters are distributed among the teams. If all the people named Peter are in one team, then Peter has a much lower chance of being on the winning team than if each team has at least one Peter in it (in which case there’s a 100 % chance that a Peter was on the winning team.)

If we know how to compute with conditional probability, these complications don’t actually make the problem harder, only more annoying. This is because we can compute the probability of the Peters being distributed various ways2² What is the probability that no Peter competed? What is the probability that exactly one Peter competed and was on the winning team, etc. and then combine the results.

However, there’s an easier way to do it by thinking combinatorically3³ And this is the reason I wrote this up in the first place: I didn’t expect to be able to use the enumerative combinatorics I learned a few months ago, but thanks to spaced repetition it was still fairly accessible in my brain!: There are \({19 \choose 5}\) ways to select the members of the winning team from all 19 colleagues. There are \({15 \choose 5}\) ways to select the winning team without picking any Peter4⁴ If we restrict the valid candidate pool to only those whose names are not Peter, there are 15 colleagues to choose from.. Thus, the probability that Peter is in the winning team is

\[1 - {15 \choose 5} \bigg/ {19 \choose 5} = 74 \%.\]

It was a fairly good guess!

Motivation: Picking the Winning Team First

We seemingly ignored a bunch of information in that calculation. In fact, the only things we did pay attention to were

• Number of colleagues
• Number of colleagues not named Peter
• Team size

What reason do we have to ignore everything else? It rests on a few assumptions. The most important of which is that the winning team is – conditional on the knowledge we have about the situation, which is very little – independent on who is actually on the team. That independence means we can pretend that a winning team was picked first, and only then was participants randomly assigned to the team.5⁵ This sort of feels like we’re reasoning with time flowing backwards, which is not a bad intuition. Stochastic independence between events means their relative location in time becomes meaningless, and in fact, time itself ceases to be a meaningful concept. At some point I want to write about how time and causality are a phenomena of logical/stochastic dependence but this is not that day.

Since we don’t have any reason to suppose that the Peters (or non-Peters) as a group were particularly likely to not be present at the competition at all, we can also reason as if every colleague has equal probability to end up in any team – including the winning team.6⁶ This sort of thinking applies to many competitive situations, which unfortunately has the side effect of making those competitions rather uninteresting.

Conditional Probability of Peter winning

When I had guessed, my wife rolled her eyes and said “I was.”7⁷ She may also have made a comment about me being boring. Now we’re in a position with more information: how confident should we be now that Peter is on the winning team? The reasoning is very similar, except now there are only four places left to fill on the winning team, since we know the identity of one of the people in it, and we have one fewer colleague to put there.

\[1 - {14 \choose 4} \bigg/ {18 \choose 4} = 67 \%.\]

This is still a good chance, but it turns out Peter was not on the winning team. I should probably have guessed “You!” after all.