Lent 2017 JRN MATHEMATICAL TRIPOS: PART IA PROBABILITY Example Sheet 1 (of 4) 1. Four mice are chosen (without replacement) from a litter, two of which are white. The probability that both white mice are chosen is twice the probability that neither is chosen. How many mice are there in the litter? 2. A table-tennis championship for 2" players is organized as a knock-out tournament with n rounds, the last round being the final. Two players are chosen at random. Calculate the probability that they meet: (a) in the first round, (b) in the final, (c) in any round. [Hint. The same probability space can be used for all three calculations.] 3. A full deck of 52 cards is divided in half at random. Find an expression for the probability that each half contains the same number of red and black cards. Evaluate this expression as a decimal expansion. Use Stirling's formula to find an approximation for the same probability and evaluate this approximation as a decimal expansion. 4. State what it means for F to be a o-algebra and for P to be a probability measure. Let (An : n E N) be a sequence of events in a probability space (2, F, P). Show, starting from the definitions, the following properties: (a) Ø and A1 U A2 and Nm-1An are events, (b) P(Ø) = 0 and P(A) = 1 - P(A1), (c) if A1 and A2 are disjoint, then P(A1 U A2) = P(A1) + P(A2), (d) if A1 C A2, then P(A1) ? P(A2), (e) P(A1 U A2) = P(A1) + P(A2) - P(A1 n A2), (f) if An [ An+1 for all n, then P(An) -> P (Un An). 5. (a) Show that, for any three events A, B, C, P(Ac n (BUC)) = P(B) + P(C) -P(BnC) - P(CnA) -P(AnB) +P(AnBnC). (b) How many of the numbers 1, ... , 500 are not divisible by 7 but are divisible by 3 or 5? 6. Let (An : n E N) be a sequence of events in some probability space (2, F, P). Set A = {W En : WE An infinitely often}, B = {WE ? : WE An for all sufficiently large n}. (a) Show that B = n=1 So TK=n Ak. (b) Show that A is an event and that R(A) P n=1 00 (c) Suppose that the series 2 P(Ak) for all n. Pk=n 00 nverges. Show that P(A) = 0. 1
7. A committee of size r is chosen at random from a set of n people. Calculate the probability that m given people will all be on the committee (a) directly, (b) using the inclusion-exclusion formula. Deduce that m 0 00 n - m X m n - j = j=0(-1)j 0 . r - m j r 8. Examination candidates are graded into four classes known conventionally as I, II-1, II-2 and III, with probabilities 1/8, 2/8, 3/8 and 2/8 respectively. Candidates who misread the rubric, a common event