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STAT1301 (T1 ans)

STAT1301 (T1 ans) - THE UNIVERSITY OF HONG KONG DEPARTMENT...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTURIAL SCIENCE STAT 1301 PROBABILITYAND STATISTICS I EXA1V.[PLE CLASS 1 1. It is given that P(A) = 0.3, P(B) = 0.2 and P(AﬂB’) = 0.24. (a) A Find P(AmB). 0‘0 é (b) Find P(AUB). '044' (c) Are events A and B mutually exclusive? Why? NO “X VCH 0624‘0 2. The menu for breakfast of a coffee shop consists‘of the following items under categories A, B and C. A B ' - C sausage noodle ' coffee egg spaghetti tea vermicelli juice porridge #% gag (a) How many different sets of breakfast can be formed? =2 xéfx}._ 2 S£ (b) If a waiter gives one randomly formed set of breakfast to a cus omer, what is the probability that (i) the drink IS coffee? (ii) spaghetti lS excluded? ' I ’ 3. A product code of a manufacturer consists of 2 letters followed by 3 digits. The manufacturer only uses the letters A, B, C and D. For example, AA032 and BD369 are valid product codes. If a product is selected at random, what is the probability that (a) it begins with the letter A? (b) the first two letters are identical? (c) the letters are identical and the digits are equal? (d) the letters are formed by B and D, and the 3- -digit number IS even? a) may WA A) z; 5) ”CHI 2 CHI”) (Mme/Q): /X¢:/ q [jg/W Kitty/x1764 [ﬁfe/5 M 63x41 ”(2570): 2% 7((0 7%“: 2E5}; KW » l _ gwﬂMQ,eut/L3 itm;%i<¢?<'i 4. 3 )boys and 4 girls s7line up randomly in a row.‘ at is t e probability that (a) all boys are on one side? (b) boys and girls are in alternate positions? . matte, ‘ (c) no 2boys Slt next to each other? i C 2 2 ’38 W riﬂed try: an ( an): iii-weir? dang/335 L)” (Mm? MKM) 3&4 a. . _ ~- _ .. 0) W02 at“ W”) [ti-("3 219: Z ' a, chase. 7% 3; ’7 Titan/f ,___L ”6* 5. Jeremy climbs along a cubical framework from a comerA to meet Davis at the opposite corner B. The framework, shown in the following ﬁgure, is formed by joining bars“ of equal length. Jeremy chooses randomly a path of the shortest length to meet Davis. An example of such a path, which can be denoted by Right— Up~F0rward~ Upakight—Forward—Forward, B is also shown in the ﬁgure. ' I 11-. Alla-l ‘ IlIIIlII AIIIIIIIV AIIIIIIII 'Ig‘__—I AIIII (a) Find the number of shortest paths fromA to B. #0f (51/95196le M} A 3 ’ 2 P 2' (b) Ifthere 1s a trap at the centre C of the framework which catches anyone assin through 4st, (i) ﬁnd the number of shortest paths ﬁom A to C, ofwsm (ii) hence, ﬁnd the probability that Jeremy will be caught{{ bygf the trap on his ways 2to B Magi/t M @221??? 6. Nadal and Federer and six other players take part in a tennis knock-out tournament. The winner of each match can proceed to the next round as shown in the following ﬁgure and the loser is knocked out. The players are randomly assigned to the eight positions in the ﬁrst round. Suppose the eight players are equally skilful. 3f Champion ................ Third round ......... Second round -4----First round (a) What is the probability that Nadal will play Federer in the ﬁrst round? (b) What is the probability that Nadal will ever play Federer in a match during the tournament? A) WM mat {aaeﬂw):~_’7 ‘ 5) Whateﬂwzn m7 WM}??? ifxgrixi/arét/(f; 81\$ 12%,: :1 /‘ ma ...
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