“I got a rock” and a probability question along with it (part 2)

In part 1 of this article, I detailed one approach for solving a simple-sounding (but still tricky!) probability question. The question was as follows:

Recap: Six rocks are put into three buckets at random, with no bucket left empty. Find the probability that at least one bucket has exactly one rock in it.

The correct answer happened to be 7/9, as you might recall. The idea was to break the possible situations into cases, but when you do this, you have to be careful. If you’re not, you may end up committing the balance fallacy. The balance fallacy is where someone erroneously assumes that a bunch of outcomes are all equally likely to occur just because they can all be listed out in a certain way.

Here’s a simple example of the balance fallacy: you flip two coins. There are three outcomes: heads and heads; heads and tails; or tails and tails. Thus each of these outcomes has the same probability of happening, 1/3.

This is a common mistake in even simple probability questions. The balance fallacy is often committed because the outcomes that are listed aren’t described clearly enough. For instance, if one coin is heads and one is tails, it very much matters which one is which — there are really two different outcomes disguised as one: where the first coin is heads and the second coin is tails, and vice-versa.

Now, let’s look at an incorrect solution to the rocks and buckets problem that commits the…