Ask for the chance of a US recession starting in 2027 and markets will tell you: about 31%. Ask a sharper question, what is the chance of a recession if the Fed cuts in December, and you get two numbers instead of one: 36% if it cuts, 23% if it does not.
Those two numbers carry more than the headline. They tell you the markets are linked, by how much, and what will happen to one the moment the other resolves. Drag the top branch and watch the rest of the tree follow.
The rule underneath is the law of total probability: the overall chance is an average of the conditional chances, weighted by how likely each branch is. For the recession market, 0.36 × 0.62 + 0.23 × 0.38 comes to about 0.31. That is not a coincidence. It is how the numbers have to fit together if the markets are consistent.
02Area is probability
Trees are good at showing paths. Grids are better at showing size. Here are the same hundred worlds as Figure 1, drawn as squares, one per world.
31 of 100 squares are filled, and 22 of them sit in the cut group. Shuffle the grid: the count stays the same, only the story gets harder to see.
Notice that most of the filled squares sit in the cut group. That does not mean cuts cause recessions. Usually it runs the other way: a weakening economy makes a cut more likely. The grid does not care about direction. It just counts.
03Chains get unlikely fast
Conditional thinking also protects you from a classic trap. A story with five steps, each likely on its own, feels convincing. Multiply the steps and it rarely is.
Each step at 70% sounds likely. All 5 together: 17%. The whole story is more likely false than true.
Psychologists call the mistake the conjunction fallacy. Every detail added to a scenario makes it more vivid and less probable. When someone walks you through a chain of events, multiply the links before you agree with the ending.
04Moving together is not causing
When two markets move together, it is tempting to read one as the cause of the other. Often both are reacting to something else. Here are sixty days of daily moves in two markets. Then flip the switch.
The two markets tend to move in the same direction on the same day: when cut odds fall, so does the S&P market. The link looks strong. Is one market moving the other?
On oil news days, the Fed market and the stock market lurch together. Take those days out and the link nearly vanishes. Neither market was driving the other; both were listening to the same news.
That is why our if-then explorer labels every pair by where its numbers come from, paired markets or our model, and why we never call a link a cause.
Conditional odds are sample data. Figure 4 is simulated. Nothing here is investment advice.