STAT 230 Exam 1 with Solution, Spring 2011

American University of Beirut 3. Chips are drawn at random, one at a time and STAT 230 without replacement from an urn that contains Introduction to Probability and Random Variables 5 pink chips and 7 white chips, until only those Spring 2011 of the same color are left. Find the probability that exactly 7 draws are needed. quiz # 1 Name: solu bin (_aw_] or q Pay P | ID: Section: 12:30 PM 2PM + G C C4 . : Cc C gy *F 1. In how many ways can three different prizes be 4 _ >? ee —3z + 6 awarded among nine contestants if no person ey C 6 is to receive more than one prize ? Ax8xF 2 ate = 504 2 A, Four couples are to be seated in a row. Find 2. One of the five élévators in a building starts the probability that each husband sits next to with seven passengers and stops at nine floors. his wife. Assuming that it is equally likely that a pas- senger gets off at any of these nine floors, find let C 2 CG, 5) Gq 2 G be the probability that at least two of these pas- te 4 tou thes sengers will get off at the same floor. A = ; ab fees} twe pesserges ger off ek be Some floor } gix2 J | ios P(A) = l- P(A) 4! : pewmlatin of He cl (<2 3] ¢% 9x8 x dnb Sx ae3 G@ cudes 18 4 et cel 3 2 : pamaia a o} ecc ( p coute xu - } ss at = 0.96 3! : bokef nb. ef pemelile 5. Let A,B and C three independent events with 7. An absentminded professor wrote 15 letters and probabilities 1/2,1/6,1/4. Find P((AN B)U C). P(& 0B) uc)= ARAB) + Plc) MArBnc) ied LR) PCB) + Plc) - HA) PIB) PO) 1,I,4 = jefe ‘ ~ 2 6 4 27. 9 - 9 SbdS = y a 14 . 6. The pdf of a random variable X is given in the following table cig | 2 1 4 P(X=2)| 1/8 2/8 5/8 Find E(2X — 1) sealed them in envelopes before writing the ad- dresses on the envelopes. Then he wrote the m ! ¢ addresses on the envelopes at random. What is the probability that no letter was addressed correctly ? | aching forme la w iHrouk reploceme A. | ne feller wes 294. crreellyf P(m) = 1- PA) | er tate isi 8. A computer is instructed to generate a random sequence using the digits 0 through 9; repeti- tions are allowed. Find the shortest length the sequence can be and still have at least 70% probability of containing at least one 4. Lek n be He lena Hh o} Lhe sequre Ae event os el con bashing al Je ash one aS Ak) 2 I- PIA)= I- (7)" IB a 9. In a squash tournament between three play- 10. ers A, B and C, each player plays the oth- A plays B, A plays C and B plays C). Assume the following probabilities: P(A beats B) = 0.6; P(A beats C) = 0.7; P(B beats C) = 0.6. Assuming independence of the match results, calculate the probability ers once (ie. that A wins at least as many games as any other player. W -{ event A wias al hesk es mend grees as “4 olta bps ib means Lest Ss Plw) = P Plass)n@ £9) PALB)a B90 € $y) 4 PALIo (c48)0 RSA) 5.2 x0.6 + 0.bro$ xo- 3 — ine. + 0.964 vo. 4 - ole olo§ + 0.112 - 0.64 Let A and B be two events with P(B) = 0.7 and P(A|B) = 0.4. Find P(AU B). P(AuB)- PIA+ p(B) — Plan®) P(A) +P 8) - (A) _Pisos)} pla) + Plans) _ p(B) « pals) x P68) = ost 0.6 x04 =0.4d 11. Ona multiple-choice exam with four choices for 12. Marwan, Jad and Kate simultaneously toss coins. each question, a student either knows the an- swer to a question or marks it at random. Ifthe probability that he or she knows the answers is 2/3, what is the probability that an answer that was marked correctly was not marked ran- domly ? R vestian answered 53 | astin nee > eel P(CIR) x PER) + PC (B) «tl ®) ims) “_—~ Aa pages cole [x & 2 g a ~ 1, t tx 2 a a ad The coin tossed by each one turns up head with probability 1/2, 1/4 and 1/3 respectively. If one person gets an outcome different from those of the other two, then he is the odd man out. If there is no odd man out, the players flip again and continue to do so until they get an odd man out. a) Find the probability that Marwan is out in the first round. l m 1.3.2, hades = 9°G*3 424 3 - +4 ~ & 2h - =f 24 b) What is the probability that Marwan will be the odd man out ? 1. = | event florwen ovt ab He 4 i round § P ( farwen ovt) = P(11,) + PI 1a) + Pig) + +> 1) 24] dy Zu] 24 2 Ore on ria YP ere ees ee 9 ons (our>