Conditionally Dependent Probabilities
For a sequence of events, multiply the probability of the first by each subsequent event’s probability conditional on earlier successes. The conditional estimates need a stated basis; the mere fact that tasks occur in order does not determine them.
THE IDEA
Overview
For events that must all occur, use the probability of each event conditional on the preceding events. These probabilities may differ from the events’ standalone probabilities.
The Safecracker Problem
Imagine a thief must open five doors, each with a combination lock. The thief has succeeded 9 out of 10 times in the past. What is the probability of opening all five doors?
The thief can reach each door only after opening the preceding ones. That access requirement alone does not show whether success at one lock changes the chance of success at another. The calculation below assumes that each success raises the estimated rate by adding it to the earlier record.
Calculating Step by Step
- Probability of opening first door (P1): 9/10 = 0.9
- P2 = P1 × (10/11) = 0.9 × 0.909 = 0.82 (after first success, batting average becomes 10 out of 11)
- P3 = 0.82 × (11/12) = 0.75
- P4 = 0.75 × (12/13) = 0.69
- P5 = 0.69 × (13/14) = 0.64
Under that updating rule, the estimated chance of opening every door is about 0.64. The historical success rate and the sequence alone do not establish that this rule is the appropriate probability model.
Key Distinction
- Mutually exclusive events (probability of A or B): Add probabilities
- Joint events (probability of A and then B): Multiply the probability of A by the probability of B given A. Independent events use the same rule, with no change in B’s probability.
Why This Matters
A project can have a much smaller chance of complete success than any individual stage. State the conditional estimates and their assumptions before combining them.
When to use it
Estimating the probability of winning a multi-stage tournament or series of games.
Assessing the likelihood of successfully completing a multi-step project or product launch.
Evaluating the chance of a patient recovering after a series of medical procedures.
Estimating sales-call conversions when earlier results may change later chances through learning or reputation.
FROM THEORY TO EVERYDAY LIFE
In practice
Safecracker Problem
A thief attempts five doors in sequence. Starting from a record of 9/10, the supplied calculation updates the estimated success rate after each assumed success and obtains about 0.64 for opening every door. That result depends on the chosen updating model.
Clinical Trial Success
A new drug must pass three phases of trials. If the probability of passing Phase I is 70%, Phase II given Phase I success is 50%, and Phase III given previous success is 80%, the overall probability is 0.7 × 0.5 × 0.8 = 0.28.
A MOMENT TO REFLECT
Put it to work.
Think of a real-world goal that requires several dependent steps to succeed. How would you calculate the overall probability?
Why do people often overestimate the likelihood of a string of successes? Can you recall a time when you made such an error?
How does the concept of updating probabilities after each event relate to Bayesian reasoning?
Often confused with
CONTINUE THE CONVERSATION
Source texts
- The Thinkers Toolkit by Morgan D Jones