AdvancedTopic 78 of 104Engineering Mathematics
Conditional probability and Bayes
P(A|B)=P(B|A)P(A)/P(B); updates belief with evidence, used in diagnostics and filtering.
In depth
P(A given B)=P(A∩B)/P(B). Bayes' theorem P(A given B)=P(B given A)P(A)/P(B) updates belief with evidence. The law of total probability sums over mutually exclusive causes. It is used in fault diagnosis, spam filters and sensor fusion.
Key points
- Prior, likelihood, posterior
- Base rates matter greatly
- Conditioning changes the sample space
ExampleA test 99% accurate for a 1-in-1000 fault still yields mostly false alarms when positive.
Bayes' rule updates a belief when new evidence arrives, which is central to diagnosis, fault detection and tracking.
What is Conditional probability and Bayes?
P(A|B)=P(B|A)P(A)/P(B); updates belief with evidence, used in diagnostics and filtering.
Why is it important to understand conditional probability and Bayes?
Bayes' rule updates a belief when new evidence arrives, which is central to diagnosis, fault detection and tracking.
Why does a positive test not always mean a high chance of disease?
If the condition is rare, most positive results are false positives. The prior probability has to be included in the calculation.
What is the prior and the posterior?
The prior is the belief before the evidence. The posterior is the updated belief after the evidence, computed with Bayes' rule.
How is Conditional probability and Bayes used in practice?
Typical applications include estimating the probability a machine is faulty after an alarm, updating a position estimate from a noisy GPS reading and interpreting a positive result from a test with a known false-positive rate.
What should I learn before Conditional probability and Bayes?
Study "Probability basics" first (topic 77 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Conditional probability and Bayes?
Prior, likelihood, posterior. Base rates matter greatly. Conditioning changes the sample space.
Can you give an example of Conditional probability and Bayes?
A test 99% accurate for a 1-in-1000 fault still yields mostly false alarms when positive.
What comes after Conditional probability and Bayes?
The next topic is "Random variables", which builds directly on this one.
How difficult is Conditional probability and Bayes?
It is classed as Advanced in Engineering Mathematics (topic 78 of 104). It is an advanced topic. It assumes you are comfortable with the intermediate topics in Engineering Mathematics.