Probability and Statistics
Core ideas
Probability rules (addition, multiplication, conditional), independence, permutations and combinations, discrete and continuous distributions, expectation and variance, binomial and normal distributions, and hypothesis testing. Understanding the difference between mutually exclusive and independent events is fundamental.
Traps
- Mutually exclusive events cannot be independent (unless one has probability zero).
- P(A|B) is not the same as P(B|A); use Bayes' theorem carefully.
- In binomial settings, verify fixed n, constant p, and independence.
- When using the normal approximation to the binomial, apply a continuity correction.
- Distinguish nPr (order matters) from nCr (order does not).
Rule of thumb
For 'at least one' problems, use 1 - P(none). For conditional probability, restrict the sample space to the condition: P(A|B) = P(A ∩ B)/P(B).
Worked example
A bag has 5 red and 3 blue balls. Two are drawn without replacement. Find P(second is red | first is blue). After a blue is removed, 5 red and 2 blue remain, so P = 5/7. Alternatively, P(second red | first blue) = P(blue then red)/P(first blue) = (3/8 · 5/7)/(3/8) = 5/7, consistent.