# mid1sol - AMSSO7 Introduction to Probability MIDTERM I Fall...

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Unformatted text preview: AMSSO7 Introduction to Probability MIDTERM I, Fall 2011 Last Name: W First Name: N ID: Show all your work for full credit H :l-€ 4. (10pts) Consider the experiment of rolling n balanced dice. Let N be the number of dice that show either a 3 or a 6. Compute the mean and variance of N. Snafu}? F’Wh :2 —% 243* NA, BOA (‘Wzél , v n - Jain}- tinge? U’MUD' n 3, ' 'q n 7 j J 5. (10pts) An urn contains 5 red and 10 blue balls. Balls are drawn sequentially from the urn Without replacement. Let X be the number of draws necessary in order to obtain exactly 4 red balls. Find the probability mass function of X. M mam: ’- 6 ~-~-—----’-- *3 ( 12v: 6. (10pts) A coin is randomly picked from a collection of 10 coins, the ith coin having a probability % of coming up heads. The coin is then ﬂipped repeatedly until a head appears. Let X be the number of ﬂips necessary. Find the probability mass function of X. [)(x: : 5: W X: [a] FHA 032m {5 Salami)? (my (gun. Sfjegw) H W — govt k4 ‘1 \m l. [0 [O [€14.2f,» 7. A random number N of dice is thrown. Assume that P(N = z for i = 1,2, 3, . . ., and let S be the sum of the scores obtained. Find the probability that S 2 4 given N is even. Stir ,U ) ’ W WWW”)“#2)”?wa Meir) - WW4) r _. , P(N‘Jévw) ’ WNW) AWL .J’r‘ 3 (9‘ [fl—é!" , ff‘f-i’fiffll'gi' 8. (10pts) A and B roll a pair of dice in turn, with A rolling ﬁrst. A’s objective is to obtain a sum of 6, and B’s is to obtain a sum of 7. The game ends when either player reaches his/ her objective, and the player is declared the winner. Find the probability that A is the winner. lit/Ell was} SA’;¢U+W\1°'§'M W? M was} sﬁagewmo} in ﬁrst mi Lug m pm): péAtﬁA:6)*P<SA:€>+ P(A\SA\$6,35=7)=P(Siité,35=7) ~+p<AIwaswiwwnggw) : + 3‘ Zéxé , \$0 9. (20pts) A man has 5 coins, two are double-headed, one is double—tailed, and two are normal (fair) coins. He shuts his eyes, picks a coin at random, and tosses it (clearly deﬁne the events you use m solving the problem)- | ; g X A) 7 i : §nwwe } (a) What’s the probability that the lower face (the side facing the ground) is a head? Lwéw‘ww at - aw a“ rm % rim m Pm: H): Wham/mm.)+1>(L.;H[A))1>M») _: {‘JS»+O + +Pd—F‘H ‘ .- 3, 3 ' 5 hat thevcoin is showing heads. What is the probability that the lower face (b) He opens his eyes and sees t ‘ z"th cl} m Wye 5Qon [a 4m gm M3. isahead? U‘: l - ’ gr (MPH fly) 3 My) ; VLAth-H)e WPWM) H pﬁ t 5. v 2 l (0) He shuts his eyes and tosses the coin again. What is the probability that the lower face is a head? “ L7. == can-teem 0; m 1mm 3;“; (M +64 Sea-ML +1531 : 3 : (L23H1U15H) {Hashim Lﬂ—“HFM,H) {l ,3, ii 7— ;i 2 t _ TE ! ' 3 .4 l '5 (d) He opens his eyes and sees that the coin is showing heads. What is the probability that the lower face is a head? PCA! {Hit-H; : E(UFH,‘M>=H Mary/M P(MI=H, mm) 2. M : 5 3'.- .L.E_ "5+4 5' (e) He discards this coin, picks another at random and tosses it. What is the probability that it shows ’- 6’ n. hem? E: Nu mm (ﬁscal/«Aw *9 u'm El VQH) ; PM Matti/04* P(H[EL)~}?CEL)+P(H‘53)‘P(—}) ’- ‘ _ , ' s‘li"t+%~%]+0+%ffé+it] ...
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mid1sol - AMSSO7 Introduction to Probability MIDTERM I Fall...

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