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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*
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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)
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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 ) ’
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3 (9‘ [fl—é!" , ff‘fi’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}
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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 doubleheaded, 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 ‘
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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. == canteem 0; m 1mm 3;“; (M +64 SeaML +1531 : 3 : (L23H1U15H) {Hashim Lﬂ—“HFM,H) {l ,3,
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(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! {HitH; : E(UFH,‘M>=H Mary/M P(MI=H, mm)
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"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(—})
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 Fall '08
 Feinberg,E
 Probability theory, Randomness, Dice, Coin, Probability mass function, lower face

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