Probability and Expected Value
Probability quantifies uncertainty and supplies the expected value used in decision-making under risk. The exam tests the classical definition, addition and multiplication rules, conditional probability and independence, Bayes' theorem in simple form, random variables, expectation and variance of a random variable, and the binomial distribution as an application.
Core rules
- Classical probability = favourable outcomes / total equally likely outcomes.
- Addition: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for mutually exclusive events the last term is zero.
- Multiplication: P(A ∩ B) = P(A) × P(B|A); for independent events P(A ∩ B) = P(A)P(B).
- Complement: P(A′) = 1 − P(A).
- Expected value E(X) = Σ x·P(x); variance = E(X²) − [E(X)]².
- Binomial: P(r) = nCr pʳ qⁿ⁻ʳ, with mean np and variance npq.
Worked example
A die is rolled. You receive ₹6 if a six appears and pay ₹1 otherwise. E(X) = (1/6)(6) + (5/6)(−1) = 1 − 0.833 = ₹0.167. The game is favourable by about 17 paise per play. Note that the ₹1 loss must be entered as a negative outcome; ignoring the losing branch would wrongly give an expectation of ₹1.
Traps
Confusing mutually exclusive with independent - they are different conditions and use different rules; forgetting to change the denominator when sampling without replacement; omitting the cost of playing from the expected value; and reporting a probability outside 0 to 1.