Events and iheir Pr<>babi,in (c) Do these results o JM^^^^ (d) Do these results ho ^ A, ,east ,w0 are alike, and it is an evens chance lhe,hirdisaheadora,,l.Th...... thai ,he third ts a hea o ^ ^ ^ their sum _ ? |s independem of ^ ^ 9. Two fair dice are rolieu. ai bv the first die. _____--.-----------_____ 1.7 Exercises. Worked examples i Th,re are two roads from A to B and two roads from B to C. Each of the four roads is blocked by mow with probability p, independently of the others. Find the probabtl.ty that there is an open road from A to B given that there is no open route from A to C. If, in addition, there is a direct road from A to C, this road being blocked with probability p independently of the others, find the required conditional probability. 2. Calculate the probability that a hand of 13 cards dealt from a normal shuffled pack of 52 contains exactly wo kings and one ace. What is the probability that it contains exactly one ace given that it contains exactly two kings? 3. A symmetric random walk takes place on the integers 0,1,2.....N with absorbing barriers at 0 and N, starting at k. Show that the probability that the walk is never absorbed is zero. 4. The so-called 'sure thing principle" asserts that if you prefer x to y given C, and also prefer x to y given Cc, then you surely prefer x to y. Agreed? 5. A pack contains m cards, labelled 1,2.....m. The cards are dealt out in a random order, one by one. Gnen that die label of the fcth card dealt is the largest of the first k cards dealt, what is the probability that it is also the largest in the pack'' Problems Kxercises [I.8.4H 1.8.14] 4. Describe the underlying probability spaces for the following experiments- (a) a biased coin is tossed three times; (b) two bulls are drawn without replacement frnm-,n ,,rn i, ■ ■ » and two vermilion balls: Ch ^ C°n,a'ncd ,wo u,,ramarine (c) a biased coin is tossed repeatedly until a head turns up. 5. Show that the probability that exactly one of the events A and B occurs is PM) + ?(B) - 2?M O B). 6. Prove that P(-4 U B U C) = 1 - ?MC | Bc n CC)?(BC | CC)?(CC). 7. (a) If A is independent of itself, show that ?{A) is 0 or 1. (b) If P(A) is 0 or I, show that A is independent of all events B. 8. Let .fbe a cr-field of subsets of Q. and suppose ? : F[0. 1] satisfies: (i)?(2) = Land fii) ? is additive, in that P(A U B) = P(A) + ?(B) whenever A n B = Z. Show that ?<2> =0. 9. Suppose (fi, F, P) is a probability space and B e JT satisfies ?(B) > 0. Let Q : F — 10. II be defined by QL4) = ¥(A | B). Show that (fi, F. Q) is a probability space. If C € Fznd Q(C) > 0. show that Q(A | C) = ?(A | B n C); discuss. 10. Let B\, B2____be a partition of the sample space U. each B, having positiv- probability, and show that P(A) = __F(/4 I Bj)?(Bj). 11. Prove Boole's inequalities: 12. Prove that 1.8 Problems ft) bo* numbers are odd? W the sum of the scores is 4-> (d> lhesumofihescor«UH- ■ 2 A fa esisdlVIS'Weby 3? 2 ^^fecuj^13',^iS 'he Pr°bability lhal °" 'he nth throw: :thenu^ofhead;:;1" (C) e«c,l, ,w0 head ^ 10 da* are equal? 3- LeiT-j-. P^edtodate? "») Ut'i-y^ in * ^0^1 nSHr cll"*d under countable intersections; that ,C) Sbo* that fitj^ C°,,eC,i0n of s«bse„ of o i • , • „ ,.fiel<» ?U collection Qf subsetstSJfnn 'yng ,„ both ^and ». Show that *« « » ^ is. »f •ts of rj ivin)l in -8l|lJW ,Pnr n __i____..««itrily a " p(f]Ai)=TnAi)-^nAiuAJ)+ £ ?