Probability space
A triple (Ω,ℱ,P): outcomes, a σ-algebra of events, and a countably additive measure with P(Ω)=1.
Find a concept
REFERENCE SHEET
Events, conditioning & Bayes
Update probabilities by restricting the sample space and renormalizing.
Before this Sets & maps
Definitions
Conditional probability
For P(B)>0, P(A|B)=P(A∩B)/P(B).
Independence
A and B are independent if P(A∩B)=P(A)P(B). A family is mutually independent when the product rule holds for every finite subfamily.
Key theorems
Total probability
For a countable partition {Bᵢ} with positive probabilities, P(A)=ΣᵢP(A|Bᵢ)P(Bᵢ).
Bayes rule
The displayed identity follows by writing the joint probability in two ways; the denominator can be expanded by total probability.
Notable applications
Classification
Combine prior class probabilities with evidence likelihoods.
Reliability
Break a system into operating scenarios and combine conditional failure probabilities.
Work it out
Try before revealing01 / Compute
A factory uses machine A for 60% of parts and B for 40%. Defect rates are 1% and 3%. Given a defective part, what is the probability it came from B?
Hint
Compute the total defect rate first.
Solution
P(defect)=0.6·0.01+0.4·0.03=0.018. Thus P(B|defect)=0.012/0.018=2/3.
02 / Explain
For a fair die, are “even” and “greater than 3” independent?
Hint
Compare the joint probability with the product.
Solution
The joint event is {4,6}, with probability 1/3. The product is (1/2)(1/2)=1/4, so they are not independent.
Connect the ideas
Build on this
These sheets use this topic as a prerequisite.
Go deeper
Further reading and full-length treatments: