cheatsheet 4.1 - 4.2 - 1iFtp-‘hat is the significance of...

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Unformatted text preview: 1iFtp-‘hat is the significance of the mean of a probability‘ dismbution? Choose the correct answer below. It gives information about how the outcomes H" [t is the expected value of a discrete random variable. Determine whether the statement is due or false. [fit is false. rewrite it as a true statement. The mean of a random variable represents the "theoretical average" of a probabilitv experiment and sometimes is not a possible outcome. Choose the correct answer below. V“ True. Decide whether the random variable x is discrete or continuous. X represents the length of time it takes to get to work. Is the random variable x discrete or continuous? Choose the correct answer below. '1'" Continuous Discrete Decide whether the random variable x is discrete or conu'nuous. 1: represents the total number of die rolls reqtu'red for an individual to roll a five. Is the random variable x discrete or conu'nuous? Choose the correct answer below. V Discrete C ontinu ous Determine whether the dismbution is a discrete probability disuibution. x Pflx} fl - 0.01 l — [LEE 2 2.66 3 — [LEE 4 — ELUl Is the probabilitv dism'bution a discrete disn‘ibution’.’ Wth Choose the correct answer below. H" No. because some of the probabilities have values greater than 1 or less than D. Use the fiequencv disuibution to conscht a probabilitv distribution. The number of dogs per household in a small town Dogs El 1 2 3 4 S Households 1485 429 loll 42 29 13 1485 PEG) = F =5 [USES 21:8 The completed table is shown below. P‘Ex) 0.638 0.199 0.094 U. [ll 9 9.913 0.006 U‘I-P-U-‘IINJI—‘gfi Eh} Find the mean of the prohahflitv dism'hution. The mean of a discrete random variable is given hv the following formula. it = Z KP Ex) Each value of x is multiplied hf; its corresponding prohahilitv and the products are added. The completed prohahilitv disnih-ulion table is shown below. rounded to the nearest hundredth. x Pflx} xPflx} El- III-.583 [l 1 0.199 [LE 2 [I'fl—H- 0.15 3 0.019 [LUIS 4- 0.013 [LBS 5 0.006 CLUE The mean of the prohahflitv dismhution is shown below. |.L= = 0— [LE — [L15 — ELUIfr— [LEI'S — CLUE = [L49 Thus the mean is approximater I15. (c) Find the variance of the prohahflitv disnihution. The variance of a discrete random variable is given hv the following formula. '52: EEK—H}2PEX} x Pix) 1—11 (11—1111 rx—olatx) [1 [1.683 — [1.4-51 [1.24- [1.165 1 [1.1 E151 [1.51 [1.215 [1. [152 2 [1.[1'1'4 1.51 2.28 [1.1651 5 [1.[1151 2.51 15.5 [1.12 4 [1.015 3.51 12.52 [1.1I5 5 [1.0116 4.51 2[1.3=1- [1.122 Finally. to find the yariance of the probability disuihutiort sum the (x - p.) 2I" [11) products found in the previous step. 52: Z (x—1931191):odes—oosa—o_1119—o_12—o_1s—o_1aa=o_1aa Thus the variance of the probability disuihulion is approximately [1.8. 1 [n the game of roulette. a player can plate a $15 bet on the number 55 and have a E probability of winning. If the metal hall lands on 33. the player wins $525. Otherwise. the casino takes the player's $15. 1iWhat is the expected value of the game to the player? If you played the game 1[1[1[1 times how much would you expect to lose? 1: Pflx) x ' Pflx} $525fiwim1ing) [1.0263115 $118159 — $15 (losing) [1.9131584 — 14.:1osao I‘Cow compute E xPflx}. 13.3159—[— Isl-[10526013 — [1.29 Thus the player's expected yalue is approximately — $0.29. A binomial experiment is a probability experiment that satisfies the follots'ing conditions. 1. The experiment is repeated for a fixed number oftriala where each dial is independent of the other dials. 2. There are only two possible outcomes of interest for each dial. The outc ornes can be classified as a success (3) or as a failure (F). 3. The probability of a success HE) is the same for each dial. 4. The random variable x counts the number of successful dials. Find the mean. variance and standard deviation of the binomial dismbution with the given values ofn and p. n = 120. p = 0.4 The mean of a binomial dismbution is given bf; - ll 2 11? where n is the number of times a dial is repeated and p is the probabilitf; of success in a single dial. Find the mean bf; substituling n = 120 and p = [L4 into the formula for the mean. u=12|3 - [Lil-=43 The variance of a binomial dismbution is given bf; .l 4 D" =flpq where q is the probabilitf; of failure in a single dial (q: 1 -p). Find the variance bf; subintuIing n = 120. p = [t4 and q = or» into the formula for the variant e. 53=1ao - o4 - o.o=2as The standard deviation of a binomial dismbution is given bf; the following formula. 0': flpq Find the standard deviation bf; subintuIing n = 120. p = [L4 and q = on into the formula for the standard deviation. 5: U 120 - [L4 - on £55.4- 29% of adults sav cashews are their favorite kind of nut. You randomlv select 12 adults and a! each to name his or her favorite nut. Find the probabilitv that the number who sav cashews arr their favorite nut is (a) exactlv three. (b) at least four. and (c) at most two. If convenient use technologv to find the probabilities. Ln a binomial experiment the probabilitv of exacdv x successes in n trials is given bv the following formula. n! Pfixl= C p“o“"‘=—p"q“"‘ “ x [n-x)!x! (a) Using 11 = 12. p = I129. q = EL? 1. and x = 3. find the probabilitv that exactlv three sav cashews are their favorite nut. PG}: negroaoflovug :5 oars (b) The probabilitv that at least four sav that cashews are their favorite nut is the sum of PG). H5). H6}. . P02). However. one can also solve this problem bv using the complement rule. P(xa4)=1—P(x<4) To find the probability that x is less than four__ find the sum of Pflfll Pflll PG} and PG). 10(0): 12:::|3(0_20}'3(0_?1)12 :5 0.010410 10(1): 12e1(0_20)1(0_?1)11 :5 0.000401 10(2): 12:::2(0_20)3(0_?1)1'3 s: 0.100000 10(3): 12:3(0_20)3(0_?1)9 :5 0.240004 Now find the probability that less than four sag; eashews are their favorite nut. P111614): P(U)—P(1)—P(2)—P(3) = 0.010410—0000431—0.100000—0.240004 = 0.523531 Finalh: find the probability that at least four sag: that eashews are their favorite nut. P(x24-j = 1-Pfixi4) = 1—0.523531 =5 [Lil-7'6 (e) To find the probability that x is at most two: find the sum ofP[fl]-__ PU} and PG}. P(x£2 = P[0)—P(1)—P(2) = 0.010410—0000431—0100000 s: 0200 Ihe graph below shows the results of a su1vev of drivers who were asked to name the most annoying habit of other drivers. You randomlv select six people who participated in the su1vev and ask each one of them to name the most annoying habit of other drivers. Let x represent the number who named talking on cell phones as the most annoying habit. Complete parts (a) through (c). :1; Talking on 131111 phone 35°F:- Slaw drim in fast lane 22% Pull}: drivers who tailgat a 13% 1Veasrethru‘ugh trai'ie 13°F:- 1 1‘?'«'n Dn—ro ad pet peeve 5 Use the binomial probabilitv formula given below to find the probabilities for the possible values of x. Pei) : flcxpan—x n! _ x n-x — [II—x)!le q Find the probabilitv for x = El. :5! [oasflomfi'fl (6-D)!fl! :5 ones PUD) = Consuuct a binomial dismbulzion for the possible values of 3c rounding to the nearest thousandth. P (x) El'fllfiE" [LE 32 0.325 0.245 U. 103 0.323 0.302 Din-lh-UJIUI—iclibd h) Find the prphahilr'ts' that exactly two people wfll narue "talking on cell phones." When the driver narues talking on cell phones as the must arums-irlg hahit it is classified as a success. Ihus find P[2}__ the probability pr successes in 6 uials. Use the part (a) binomial disuihuticrn to find the probability. P(2)=t+_326 c) Find the prphahflity that at least five people will name "talking on cell phones." The prphahilr'ts' of 5 or more successes is the sum of the probability of 5 successes and the probability of IS successes. Plfixas) = Fifi—P03) = [LUBE-[LUKE Thus P(x 2 s] = ates. The following histograms each represent binomial distributions. Each dismbulion has the same number ofuials n but different probabilicl' 'es of success p. [EJPH 0.? E} 0.? E} 0. 0. 0.5 0.5 0. 0. 0. 0. 0.2 0.2 0.1 _ 0.1 _ D 01234 i‘ D 01234 3‘ '3ng I” l} 0. 0.5 0. 0. 0.2 0.1 _ D 01234 }‘ 1234 Using n=4. p= [L5. and q: [*5 obtain the following probability disuibulion. x [l 1 2 3 4 P 0.003 [L250- [L37'5 [LE'U [L063 Ihe corresponding histogram is shown below. Match p = [+5. p = [+5. p = [Ll| with the correct graph. To consuuct the binomial dismbulion. find the probability for each yalue of :1. Recall that the probability of exactly 3: successes in n uials is given by the formula below. P (x) = anp‘qn'” L'sing n =4. p= [P.El. and q = 0.7". obtain the following probability dismbu‘cion. x [l 1 2 3 4 P [l.24[l [L412 [[2155 0.030 [L008 Now. graph the probability dismbulion using a histogram as shown below. Finally using n =4__ p= [Ll and :1: [Li obtain the following prohahflity distribution. x U 1 2 3 4- P [LUBE [LU-"Jo 0.265 [L412 0.240 Graph the probability distribution. PEX} ? I: III. El. El. III. III. III. El. 01234 Thu; conclude that (a) graph corrosponds to p = [Li Eh} graph corrosponds to p = [Li and (c) graph corresponds to p = [Ll ...
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This note was uploaded on 12/30/2009 for the course MATH 1410 taught by Professor Vergo during the Spring '09 term at Metropolitan Community College- Omaha.

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cheatsheet 4.1 - 4.2 - 1iFtp-‘hat is the significance of...

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