Probability and Statistics
What the exam tests
Probability and Statistics tests counting, probability rules, conditional probability, expectation, and basic data analysis. Kyoto questions often require choosing between permutations and combinations before any probability is computed.
Core rules
- Permutations count ordered arrangements: nPr = n!/(n - r)!. Combinations count unordered selections: nCr = n!/[r!(n - r)!].
- P(A ∪ B) = P(A) + P(B) - P(A ∩ B). For independent events, P(A ∩ B) = P(A)P(B).
- Conditional probability: P(A | B) = P(A ∩ B)/P(B), provided P(B) > 0.
- Expected value of a discrete variable: E(X) = Σ x_i P(X = x_i). Variance is E(X^2) - [E(X)]^2.
- For 'at least one,' the complement is often simpler: P(at least one) = 1 - P(none).
Worked example
A bag contains 5 red and 3 blue balls. Two balls are drawn without replacement. The probability of one red and one blue is [C(5,1)C(3,1)]/C(8,2) = (5·3)/28 = 15/28. Equivalently, the ordered cases are RB and BR: (5/8)(3/7) + (3/8)(5/7) = 30/56 = 15/28.
Traps
Do not use combinations when order matters, or permutations when it does not. In conditional probability, the denominator is the condition, not the whole sample space. Without replacement changes the second probability. The expected value need not be a possible value of the variable.