Exam 3 Review (Sections Covered: , )

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1 19 Exam Review (Secions Covered: ) 1 Adieisloadedandihasbeendeerminedhaheprobabiliydisribuionassociaedwih he experimen of rolling he die and observing which number falls uppermos is given by he following: Soluion Simple Even Probabiliy {1} 18 {} 1 {} 19 {4} {5} 15 {6} 15 (a) Wha is he probabiliy of he number being even? (b) Wha is he probabiliy of he number being eiher a 1 or a 6? (c) Wha is he probabiliy of he number being less han 4? D Le S {s 1 s s s 4 s 5 s 6 } be he sample space associaed wih he experimen having he following probabiliy disribuion (Ener your answers as fracions) Oucomes s 1 s s s 4 s 5 s 6 Probabiliy 1 (a) Find he probabiliy of A {s 1 s } + Tz (b) Find he probabiliy of B {s s 4 s 5 s 6 } (c) Find he probabiliy of C S 1 z+±z+zs@ PCS ) 1 4

2 4 An experimen consiss of selecing a card a random from a 5card deck Refer o his experimen and find he probabiliy of he following evens (a) A spade or a Jack is drawn ± +54 (b) Ablackcardoraisdrawn +5 (c) Ahearorafacecardisdrawn +1 1 spades 4 Jacks and 1 Jack glz D of spades if 6 black cards heas ' ' face cards zy 4 's " sad Zubzdeg ek ad!e±p Cox 4 Le S {s 1 s s s 4 s 5 s 6 } be he sample space associaed wih an experimen having he following parial probabiliy disribuion s s 1 s s s 4 s 5 s 6 P(s) 4 9 Consider he evens: A {s 1 s s 5 } B {s s 5 s 6 } C {s 1 s s 4 s 6 }andd {s 1 s s } Calculae he following probabiliies (Give answers as fracions) (a) P (s ) (b) P (D) (c) P (B c ) (d) P (A c \ B) (e) P (C c [ D) 9f # Fa sea 'S D ea +sea+ a#z a+ea D yaskazaza Fall 016 Maya Johnson

3 PLE 5 Le E and F be wo evens of an experimen wih sample space S Suppose P (E) 059 P (F )08 and P (E c \ F )08 Calculae he probabiliies below (a) P (E c ) al ) 1 59 (b) P (E \ F ) (c) P (E [ F ) g 8 (d) P (E c \ F c ) (e) P (E [ F c ) 6 Five marbles are seleced a random wihou replacemen from a jar conaining four whie marbles and six blue marbles Find he probabiliy of he given evens (Round answer o hree decimal places) (a) Three of he marbles are whie (b) Two of he marbles are blue (c) None of he marbles are whie # 461%4945 Cl6 a e# Edenae 's Fall 016 Maya Johnson

4 7 7 From he ree diagram find he following (a) P (A \ E) ( 7 X l (b) P (A) ( I + ( ) ( X l ) D (c) P (A E) MARET Kf 8 Two machines urn ou all he producs in a facory wih he firs machine producing 75% of he produc and he second 5% The firs machine produces defecive producs 5% of he ime and he second machine 7% of he ime (a) Consruc a ree diagram from he given informaion aen? ± (b) Wha is he probabiliy ha a defecive par is produced a his facory given ha i was made on he firs machine? P ( D I Ml ) P(DpqµM FIT 4 Fall 016 Maya Johnson

5 5 4 4 (c) Wha is he probabiliy ha a defecive par is produced a his facory? P ( D) L ) ) a@ 9 Suppose A and B are wo evens of a sample space S where P (A) 08 P (B) 04 and P (A [ B) 04 (a) Wha is P (A \ B)? + p ( An B ) D 4 8 PC An B) (b) Are A and B independen evens? P ( A) P ( B) ( 8 )( P ( An B) I PLAN B) PIA ) PC B) ) a 067 so no independen! 10 If A and B are independen evens P (A) 05 and P (B) 055 find he probabiliies below (Ener answers o four decimal places) (a) P (A \ B) ( 5 )( 55 ) 19@ (b) P (A [ B) @ (c) P (A c \ B c ) ( l ) a 9@ (d) P (A c [ B c ) a (155) 95 8@ 5 Fall 016 Maya Johnson

6 11 Find P (F B) andp (E A) using he ree diagram (Round answers o hree decimal places) P ( F I B) PlFnB B) ( 7 5 ) ( Three machines urn ou all he producs in a facory wih he firs machine producing 5% of sssii@pielahreptjsyyistfosze06fs he producs he second machine 5% and he hird machine 40% The firs machine produces defecive producs 6% of he ime he second machine 17% of he ime and he hird machine 4% of he ime (a) Consruc a ree diagram from he given informaion D 4% M D M +4 µ D #m z nd (b) Wha is he probabiliy ha a nondefecive produc came from he second machine? (Round answer o four decimal places) P ( 1 A medical es has been designed o deec he presence of a cerain disease Among people who have he disease he probabiliy ha he disease will be deeced by he es is 091 However among hose who do no have he disease he probabiliy ha he es will deec he presence mundhmmmnbafi5ysn8pfxsef6t@i of he disease is 004 I is esimaed ha % of he populaion who ake his es acually have he disease (Round answers o hree decimal places) (a) Consruc a ree diagram from he given informaion 0 H #Mµ c # is D Dc 6 Fall 016 Maya Johnson

7 4 I 04 (b) If he es adminisered o an individual is posiive (he disease is deeced) wha is he probabiliy ha he person acually has he disease? P ( H I D) P ( Had ) 14 For each of he following experimens give he range for he random variable X and classify i as finie discree infinie discree or coninuous (a) X Thenumberofimesadieishrownunilaappears } & (D) ea µ X is infinie ; discree (b) X Thenumberofhoursachildwacheselevisiononagivenday X{ x lo ± 4 } ; X is coninuous (c) Cards are seleced one a a ime wihou replacemen from a wellshu ed deck of 5 cards unil a queen is drawn Le X denoe he random variable ha gives he number of cards drawn 15 Deermine which of he follow ables represen valid probabiliy disribuions of a random variable X Xz { 1 X is finie 49 } ; discree (a) x P (X x) l No valid! > 1 (b) (c) x P (X x) x P (X x) l Valid l 4 No l + valid! 1 < 1 7 Fall 016 Maya Johnson

8 16 Two cards are drawn from a wellshu ed deck of 5 playing cards Le X denoe he number of aces drawn Consruc he probabiliy disribuion of he random variable X (Round answer o hree decimal places) 17 A box has 5 yellow 7 gray and black marbles Three marbles are drawn one a a ime wihou replacemen from he box Le X be he number of gray marbles drawn Consruc he probabiliy disribuion of he random variable X (Roundanswerohreedecimalplaces) PCxaoClc4ygzTjs5ljPlxal7iCl4cYjcs48yk5eEsEmxnYsEfPlxaof8yIpiziPlxaDaChgDjGlja44PlxiHHcYjglgI69nxmeiIsionaxaeEEnI@pYxfE988O 18 A man purchased a $ year ermlife insurance policy for $90 Assuming ha he probabiliy ha he will live for anoher year is 0988 find he company s expeced ne gain ECX )< 988/90 ) 19 Amanwishesopurchasealifeinsurancepolicyhawillpayhebeneficiary$5 000 in he even az/qozoooo)$@p#?q*5ye(x)96yo4(yzsooo)o ha he man s deah occurs during he nex year Using life insurance ables he deermines ha he probabiliy ha he will live anoher year is 096 Wha is he minimum amoun ha he can expec o pay for his premium? Hin: The minimum premium occurs when he insurance company s expeced profi is zero Minimum paymen y is $# y ziooo 8 Fall 016 Maya Johnson

9 0 Find he following: (a) Given P (A) /9 find odds in favor and odds agains A occurring FI?# oodddsasnafaau (b) Given P (A) 1/50 find odds in favor and odds agains A occurring Odds in favor :lo@ Odds agains :zq+oz/f (c) Given P (A) /5 find odds in favor and odds agains A occurring Odds in favor : odds 1 Find he following: agains :z@ (a) If he odds in favor of you winning a game are 11 o 1 find he probabiliy ha you will win he game and he probabiliy ha you will lose he game PC win )# )@ ; PC lose (b) If he odds agains you winning a game (odds ha you lose) are 6 o 1 find he probabiliy ha you will win he game and he probabiliy ha you will lose he game PC Win P( lose ) ) ; Find he mean median mode sandard deviaion and variance of he probabiliy disribuion for he hisogram shown (Round answer o wo decimal places) " f mx#eswienfii$ 9 Fall 016 Maya Johnson

10 The disribuion of he number of chocolae chips (x) in a cookie is shown in he following able x P (X x) : Find he he mean median mode sandard deviaion and variance of he number of chocolae chips in a cookie (Round answers o wo decimal places) Ills%e xi± *e 4 A panel of 76 economiss was asked o predic he average unemploymen rae for he upcoming year The resuls of he survey follow 1 Unemploymen Rae % Economiss : Find he he mean median mode sandard deviaion and variance of he average unemploymen rae for he upcoming year (Round answer o wo decimal places) EI± IiIEIIi en 5 I is esimaed ha one hird of he general populaion has blood ype A+ A sample of six people is seleced a random (Round answers o four decimal places) 6 P 4 (a) Wha is he probabiliy ha exacly wo of hem have blood ype A+? P ( X ) Bin on pdf ( 6 4 ) 9 (b) Wha is he probabiliy ha a mos wo of hem have blood ype A+? P ( X ± ) z B in one df ( 6 4 ) 6@ 10 Fall 016 Maya Johnson

11 6 The manager of Toy World knows ha he probabiliy an elecronic game will be reurned o he sore is 0 If 54 games are sold in a given week deermine he probabiliies of he following evens (Round answers o four decimal places) n 54 p (a) No more han games will be reurned P ( X E R ) Binomcdf (54 ) 5@ (b) A leas 8 games will be reurned P ( 1/811 Binomcdf ( 54 (c) More han 5 games bu fewer han 14 games will be reurned P ( 5 < x < 14 ) Binomcdf ( 54 I ) Binomcdf ( 54 6@ 5) 7 Acoinisbiasedsohaheprobabiliyofossingaheadis046 If his coin is ossed 54 imes deermine he probabiliies of he following evens (Round answers o four decimal places) na 54 p 46 (a) The coin lands heads more han 1 imes p( X > 111 Binomcdf ( (b) The coin lands heads fewer han 8 imes P( X < 8 ) Binomcdf ( 54 (c) The coin lands heads a leas 0 imes bu a mos 7 imes 467) ) 8@ 7l@ P ( 0 Ex 47 ) Binomcdf ( ) Binomcdf ( ) 8 Acompanyfindshaoneouofhreeemployeeswillbelaeoworkonagivenday Ifhis company has 60 employees how many employees can hey expec o be lae o work on a given day? 695 D PN Us ) ( 60 ) opeo@ 9 Anewdrughasbeenfoundobee eciveinreaing70%ofhepeoplea icedbyacerain disease If he drug is adminisered o 500 people who have his disease wha are he mean variance and he sandard deviaion of he number of people for whom he drug can be expeced o be e ecive? (Round answers o wo decimal places) Mean a pn a )( 5095 Vary NPCI 1500 P ) ) D 0 A6sideddieisloadedsohaheprobabiliyofrollingais/7 If you roll his die 600 imes how many imes can you espec he die o land on? x E( X ) p n ( I ) ( and f fo5l0@ rounded o MEET Expec?# 11 Fall 016 Maya Johnson