Probability trees and conditional expected values

How do you lay a forecast out as a tree?

A probability tree diagram shows an uncertainty that plays out in stages. Each point where the uncertainty resolves is a node . Branches leave the node, each labeled with its probability, and the probabilities leaving one node add to 1. Probabilities in the second stage are conditional probability values: the chance of an outcome given the branch you are on.

Probability treethe first uncertainty, then the outcomes within each branch

• Node Know the idea a point where uncertainty resolves
  • Branch probabilities from one node they add to 1
• End-branch (joint) probability Use it multiply along the path
  • All end branches add to 1
• Conditional expected value Memorize the forecast within one first-stage branch
  • Unconditional E(X) weight each by its branch probability and add

$$P(A \text{ and } B) = P(B \mid A)\,P(A)$$

$$P(A \mid B) = \dfrac{P(A \text{ and } B)}{P(B)}, \quad P(B) \neq 0$$

The two formulas are the same relationship read in opposite directions. Going down a tree you multiply; coming back, you divide by the probability of the branch you are on. For example, P(growth 10% | expand) = 0.42 ÷ 0.6 = 0.7.

What is a conditional expected value?

A conditional expected value is the expected value of a variable if one particular scenario happens. You compute it exactly like an ordinary expected value, but with the probabilities inside that branch.

$$E(X \mid S) = P(x_1 \mid S)\,x_1 + P(x_2 \mid S)\,x_2 + \cdots + P(x_n \mid S)\,x_n$$

How do you combine the branches into one forecast?

Weight each conditional expected value by the probability of its scenario and add. This is the total probability rule for expected value . It works when the scenarios are mutually exclusive (only one happens) and exhaustive (together they cover everything).

$$E(X) = E(X \mid S_1)\,P(S_1) + E(X \mid S_2)\,P(S_2) + \cdots + E(X \mid S_n)\,P(S_n)$$

Move the first-stage probability and watch the two conditional expected values stay put while the overall forecast slides between them. The overall figure always lies between the branch forecasts.

The practical advantage is that an analyst can revise one branch, say a higher chance of a slowdown, without rebuilding the forecast. Only that probability and the weighted sum change.

Exam tip If your overall forecast lies outside the range of the branch forecasts, you have mis-weighted something.

What about the variance in a tree?

Variance also has a conditional version. A conditional variance measures spread inside one scenario, around that scenario’s expected value.

$$\sigma^2(X \mid S) = \sum_{i} P(x_i \mid S)\,[x_i - E(X \mid S)]^2$$

Go deeper Why the overall variance exceeds the conditional ones

The overall variance splits into two parts: the probability-weighted average of the conditional variances, plus the probability-weighted squared gaps between each conditional expected value and the overall expected value. For the retailer: 0.6 × 3.36 + 0.4 × 9.00 = 5.62, and 0.6 × (8.8 − 5.68)² + 0.4 × (1.0 − 5.68)² = 5.84 + 8.76 = 14.60. Together they give 20.22. A tree that separates good and bad scenarios clearly has a large second part.

Go deeper More than two stages

Trees can run for as many stages as the problem needs: each end node becomes a new node with its own branches. You multiply along the full path for the joint probability and fold the tree back from the right, one stage at a time. A share-price tree in the binomial option model has exactly this shape, with an up and a down branch at every node.