Reverse a probability question
Does being late mean it probably rained?
Entering midway? Revisit Follow growth one period at a time for the preceding worked example, or start here and use the explanation below.
Start with a concrete question
An original synthetic travel model supplies rain probability and different late-arrival rates for rainy and dry days. Use mutually exclusive branches rather than guessing.
Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.
What do you expect?
Predict whether a high late rate on rainy days guarantees rain was likely on a late day.
Change one thing. Inspect what happens.
- Compute rainy-and-late and dry-and-late branches before adding.
- Reduce rain probability while keeping conditional rates fixed.
- Set both late rates to zero and explain why conditioning on late becomes undefined.
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Compare the current results
Every bar starts at zero and uses the same numeric scale in this chart. The upper bound automatically adjusts to the largest current value (at least 1), so compare the numbers when changing inputs. Units and quantities come from each label; different kinds of quantity are not interchangeable. Displayed values are rounded to at most six significant digits.Follow the calculation
- Rain and late: 0.3 × 0.6 = 0.18.
- Dry and late: 0.7 × 0.1 = 0.07.
- Add mutually exclusive paths: 0.18 + 0.07 = 0.25.
- Among late arrivals, rain share = 0.18/0.25 = 0.72.
Conditioning reverses the question. Rain given lateness need not equal lateness given rain.
Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.
Explain the mechanism
Reverse a probability question
An original synthetic travel model supplies rain probability and different late-arrival rates for rainy and dry days. Use mutually exclusive branches rather than guessing.
Predict before changing inputs
Predict whether a high late rate on rainy days guarantees rain was likely on a late day.
Understand the mechanism
The word given changes the denominator. P(late given rain) looks only at rainy days; P(rain given late) looks only at late days.
At defaults, rain-and-late is 0.3 × 0.6 = 0.18; dry-and-late is 0.7 × 0.1 = 0.07. Among late days, rain share is 0.18/0.25 = 72%.
Multiplying along a branch uses a conditional probability, not an assumption that rain and lateness are independent.
Calculate it yourself
P(late)=P(rain)P(late|rain)+P(dry)P(late|dry); P(rain|late)=P(rain and late)/P(late).
Guided investigation
Compute rainy-and-late and dry-and-late branches before adding.
Reduce rain probability while keeping conditional rates fixed.
Set both late rates to zero and explain why conditioning on late becomes undefined.
Assumptions and limitations
Probabilities are invented stable rates and omit other causes.
A zero-probability conditioning event has no posterior in this finite model.
Further investigations
High school: Draw a frequency tree for 100 hypothetical journeys.
College: Derive Bayes’ rule from joint probabilities and check zero denominators.
Working professionals: Inspect a forecast decision where base rates matter more than headline accuracy.
Try a fresh case
Rain-and-late probability is 0.12 and dry-and-late probability is 0.08. What percent of late days are rainy?
This is a public practice check. Retries support learning; matching a number does not certify independent proficiency.
Take the investigation further
Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.
School investigation
Draw a frequency tree for 100 hypothetical journeys.
College investigation
Derive Bayes’ rule from joint probabilities and check zero denominators.
Workplace investigation
Inspect a forecast decision where base rates matter more than headline accuracy.
Keep the evidence, choose your next step
Completion records your own decision that you have worked through this investigation. Practice checks and independent project evidence are separate.