Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Separate prior odds from the evidential strength of an observation.
- Justify the conclusion "Posterior probability=(4/9)/(1+4/9)=4/13" using the stated assumptions.
Before you start
Conditional probability and odds.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
A machine fault has prior probability 0.1; an alert occurs with probability 0.8 under a fault and 0.2 without one. Find the posterior fault probability.
Why this math matters
Separate prior odds from the evidential strength of an observation. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The two fault hypotheses are exhaustive.
- The supplied alert rates apply to this machine population.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Update odds with a likelihood ratio
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A machine fault has prior probability 0.1; an alert occurs with probability 0.8 under a fault and 0.2 without one. Find the posterior fault probability.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
Prior odds=0.1/0.9=1/9
Odds compare fault probability with nonfault probability.
Work through the mathematics
Likelihood ratio=0.8/0.2=4; posterior odds=4/9
The alert multiplies the prior odds by its relative likelihood.
Check the conclusion
Posterior probability=(4/9)/(1+4/9)=4/13
Converting odds back to probability avoids confusing sensitivity with the posterior.
The result
Posterior probability=(4/9)/(1+4/9)=4/13
Converting odds back to probability avoids confusing sensitivity with the posterior.
Common mistakes to catch
- An alert likelihood is not the probability of a fault after an alert.
- Posterior odds must be normalized to become a probability.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What if the likelihood ratio were one?
Show a hint
Multiply prior odds by one.
Reveal answer and explanation
The posterior equals the prior
The observation does not distinguish the hypotheses.
Practice 2
If prior odds are 2:1 and the likelihood ratio is 3, what is the posterior probability?
Show a hint
Convert resulting 6:1 odds.
Reveal answer and explanation
6/7
The six favorable parts are divided by seven total parts.
Take the idea with you
Compare how the same evidence changes decisions under different baseline fault rates.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Model sampling without replacement
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