Left to right: your decision, then chance events with probabilities on every branch, then your second decision, then the outcome. Multiply along a path to get its probability; add up the paths that end in a car.
square = a decision you make circle = a chance event (branches sum to 1)■ car■ goat
Strategy: stay
1/3
1/6 + 1/6 — only the paths where your first guess was already right
Strategy: switch
2/3
1/3 + 1/3 — every path where your first guess was wrong
Why the tree makes it obvious
The whole puzzle lives in one column — Monty's branch probabilities.
When the car is behind your door, Monty has a free choice of two goat doors, so each branch gets 1/2. When the car is anywhere else, his hand is forced and that single branch gets probability 1. So seeing him open Door 3 is twice as likely in the world where the car sits behind Door 2. That 2:1 likelihood ratio is the entire answer.
The same thing as a Bayes update
You picked D1, Monty opened D3. Reading the numbers straight off the tree:
Order the timeline left to right. Every node is either something you choose (square) or something the world chooses (circle). Never mix the two in one node.
Label every branch with its probability, conditional on everything to its left. Branches leaving a circle must sum to 1.
Multiply along each path to get that path's probability. All the leaf probabilities sum to 1 — that's your arithmetic check.
Write the payoff at each leaf — win/lose, or a dollar amount.
Roll back, right to left. At a circle take the probability-weighted average; at a square take the best branch. That choice is your answer.
The assumption people miss
The 2/3 depends entirely on the branches labelled 1: Monty always opens a goat door and always offers the switch. Change that rule and those numbers change. If he opened a door at random and it merely happened to show a goat, both surviving doors sit at 1/2 and switching gains you nothing — same tree, different probabilities in Monty's column.