Lent 2015 RRW MATHEMATICAL TRIPOS: PART IA PROBABILITY Example Sheet 1 (of 4) Each sheet contains about 12 fairly straightforward 'Exercises', a few more challenging and/or lengthy 'Problems', and also a couple 'Puzzles'. Please work through the Exercises and Problems. The Puzzles are for enthusiasts, and might be fun to talk about in supervision if you have done everything else. Exercises 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" equally good players is organized as a knock-out tournament with n rounds, the last round being the final. Two players are chosen at random. Calculate proba- bilities they meet (i) in the first round, (ii) in the final, (iii) in any round. [Hint: Can the same sample space be used for all three calculations?] 3. A full deck of 52 cards is divided into half at random. Use Stirling's formula to estimate the probability that each half contains the same number of red and black cards. 4. You throw 6n dice at random. Show that the probability that each number appears exactly n times is 6n (6n)! 1 (n!)6 6 . Use Stirling's formula to show that this is approximately cn-5/2 for some constant c to be found. 5. The first axiom of probability was stated as "I. 0 < P(A) ? 1 for all A C 52". Show that equivalent axioms are ones in which this is changed to the weaker requirement: "I. P(A) ? 0 for all A C 52". 6. (i) If A, B, C are three events, show that P(Ac n (BUC)) = P(B) + P(C) - P(BnC) - P(CnA) -P(AnB)+P(AnBnC). (ii) How many of the numbers 1, ... , 500 are not divisible by 7 but are divisible by 3 or 5? 7. A coin is tossed repeatedly. The outcomes of the tosses are independent. Let Ak be the event that the kth toss is a head, k = 1, 2,. . . . 00 00 (i) Explain in words what happens when the event B = UJA% occurs. n=1 k=n (ii) Express, similarly as in (i), the event C, that 'an infinite number of heads occur'. (iii) What do you think are the probabilities of B and C when P(Ak) = p for all k and 0 < p < 1? Can you prove this? 1 === :
(iv) Use the properties of P as set out in Lecture 4 to give a rigorous calculation of P(B) when the problem is changed so that tosses are independent, but are made with coins of different biases, such that P(Ak) = Pk and Ek Pk < 00. 8. Let A1, ... , Am [ {1, ... , n} be finite sets with Ai ¢ Aj and let di = | Ai|. Let 01, ... , On be a randomly chosen