Probability Tree Analysis
A probability tree analysis is a step-by-step method for mapping out all possible scenarios in a decision problem, assigning probabilities, and calculating conditional probabilities to answer questions about events and decisions.
THE IDEA
The Six Steps for a Probability Tree
Step 1: Identify the problem.
Step 2: Identify the major decisions and events to be analyzed.
Step 3: Construct a scenario tree portraying all important alternative scenarios.
- Ensure that decisions/events at each branch are mutually exclusive.
- Ensure that decisions/events at each branch are collectively exhaustive.
Step 4: Assign conditional probabilities to the possible events following each branch point. Those probabilities must sum to 1.0. A chosen action is a decision, rather than a chance event requiring a probability.
Step 5: Multiply the conditional probabilities along each path to calculate that scenario’s joint probability.
Step 6: Calculate the answers to probability questions relating to the decisions/events.
The examples combine mutually exclusive outcomes with probabilities conditional on earlier events.
When to use it
Evaluating the accuracy of medical diagnostic tests.
Making investment decisions under uncertainty, such as whether to launch a new product.
Assessing and prioritizing risks in project management.
Choosing between marketing strategies by modeling different customer response scenarios.
FROM THEORY TO EVERYDAY LIFE
In practice
Medical Test Accuracy
Consider a test for a disease with 99% sensitivity (true positive) and 95% specificity (true negative). The disease prevalence is 1%. Using a probability tree, you can calculate the probability that a person actually has the disease given a positive test result. Step 1: Identify the problem (probability of disease given positive test). Step 2: Events: have disease vs. not have disease; test positive vs. test negative. Step 3: Build tree with two main branches (disease, no disease), each with sub-branches (positive, negative). Step 4: Assign probabilities: P(disease)=0.01, P(no disease)=0.99; P(positive|disease)=0.99, P(negative|disease)=0.01; P(positive|no disease)=0.05, P(negative|no disease)=0.95. Step 5: Calculate joint probabilities. Step 6: Answer: P(disease|positive) = (0.0099) / (0.0099 + 0.0495) ≈ 16.7%.
Business Investment Decision
A company must decide whether to invest in a new product. Two main uncertainties: market demand (high, medium, low) and competition (enter or not). Step 1: Decide whether to invest. Step 2: Events: demand levels (probabilities 0.3, 0.5, 0.2) and competition (probability 0.4 if high demand, 0.6 otherwise). Step 3: Build tree: invest branch with first demand, then competition. Step 4: Assign probabilities. Step 5: Calculate each path’s joint probability and net present value. Step 6: Weight the values by their probabilities. The example does not supply investment costs or payoffs, so a numerical expected value cannot yet be calculated.
A MOMENT TO REFLECT
Put it to work.
Think of a decision you face that involves uncertainty. How would you map out the possible branches in a probability tree?
How can you ensure that your probability tree branches are mutually exclusive and collectively exhaustive?
Given a probability tree, how would you calculate the probability of a specific combined scenario, such as both Event A and Event B occurring?
CONTINUE THE CONVERSATION
Source texts
- The Thinkers Toolkit by Morgan D Jones