The following may be of use in this test: 5! = Two rows of Pascal s triangle are:

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1 Topi 8: Proaility Pratie S Short answer tehnology- free The following may e of use in this test: 5! = Two rows of Pasal s triangle are: Name: ontainer holds 20 irular piees eah of the same size. Written on eah is a different numer, one of the integers from 1 to 20. One of the piees of paper is seleted at random. alulate the proaility the numer on the seleted pi is: a not a multiple of 3 is a prime numer is either a numer greater than 15 or smaller than 8 d is a prime numer given it is a numer greater than 15 e is even and a multiple of 3 2 For events and, Pr( ) = 0.25, Pr( ) = 0.32 and Pr( Ç ) = a Show the given proailities on a proaility tale (Karnaugh map) and omplete the entries in the tale. alulate: i Pr( È ) ii Pr( ) John Wiley Sons ustralia, Ltd 1

2 3 1 1 For two events and, Pr( ) = and Pr( ) =. 4 2 a If and are mutually exlusive events, alulate Pr( È ). If and are independent events alulate Pr( È ). If Ì alulate Pr( È ). 4 group of 3 oys and 4 girls are arranged in a row. a What is the total numer of possile arrangements? In how many of the arrangements are the 3 oys together? What is the proaility that the girls and oys are in alternate positions? 5 a How many distint arrangements of the letters of the word TOMTO are possile if the letters are arranged: i in a row John Wiley Sons ustralia, Ltd 2

3 ii in a irle? alulate the proaility that the two letters T are together when the letters of the word TOMTO are arranged in a row. 6 In how many ways an a ommittee of 3 people e hosen from a group of 2 men and 4 women a without restrition with at least one man on the ommittee? Jane is one of the 4 women. alulate the proaility that Jane is seleted for the ommittee. Multiple hoie 1 The Venn diagram shows the results of a survey of 100 people who dislosed whether they added yoghurt (Y) or milk (M) to their reakfast granola. What is the proaility that a randomly hosen person from this group did not add yogurt to their granola? ag ontains 8 green alls and 7 yellow alls. Two alls are drawn without replaement. The proaility that oth are green is:!" "# '(!! ( John Wiley Sons ustralia, Ltd 3

4 3 n uniased oin is tossed three times. The proaility of at least one Head is: " " ( ) 4 In a kithen drawer there are 10 size atteries and 8 size atteries. Four of the atteries do not work and three of the atteries do not work. One attery is hosen at random from the drawer. The proaility the hosen attery is size and works is: ' ) 5 Given Pr( ) = 0.3, Pr( ') = 0.6 and Pr( ) = 0.1, Pr( ) equals: " (! ( " (#!# 6 swimmer ompetes in two events in whih the hanes of her winning are 0.6 and 0.3 respetively. The proaility she wins exatly one of the events is: Numers are formed using the digits 9,8,7 and 6 without repetition. How many numers greater than 800 is it possile to form? There are 8 fition ooks and 12 non-fition ooks on a shelf. If 4 ooks are hosen at random, then the proaility of otaining an equal numer of fition and non-fition ooks is: "' '! *( (( "# ((!#( (( '' ' John Wiley Sons ustralia, Ltd 4

5 9 multiple hoie test has 10 questions, eah with 5 possile answers. student who has not prepared for the test guesses the answers to eah question. The proaility that the student gets either 4 or 5 orret answers is given y: (0.2) (0.8) + (0.2) (0.8) (0.8) (0.2) + 5(0.8) (0.2) (0.2) + 5(0.2) (0.8) + 5(0.8) (0.2) (0.8) + (0.2) (0.8) 10 In the expansion of 20 ( p+ q), the numer of terms and the term ontaining terms, ( 12 ) pq 20 terms, ( 20 ) pq 21terms, ( 20 ) pq 21terms, ( 20 ) pq 19 terms, ( 20 ) pq 8 p are, respetively: xtended response 1 Two friends, Jude and Lee, regularly play Srale and adminton with eah other. It is estimated that in the long run Jude wins 70 of the games of Srale and 50 of the games of adminton. a On a day when Jude and Lee play Srale in the morning and adminton in the afternoon, alulate the proaility that: i Jude wins oth games ii Jude loses oth games iii Lee wins one of the games ut not oth. In the evening Jude deides to arry out a simulation of the situation using a random numer tale and a oin as follows: The digits 0,1, 2 are assoiated with Lee winning Srale and the digits 3, 4,5, 6, 7,8,9 are assoiated with Jude winning Srale. Head on the oin is assoiated with Jude winning adminton and a Tail with Lee winning adminton. xplain why this is an appropriate model for the results of the two games. The results of 20 trials are 2H 4H 2T 6T 4T 6T 6T 5H 0H 7H shown. 1H 4H 3T 3T 9H 3T 5T 5H 8T 6T John Wiley Sons ustralia, Ltd 5

6 What does the first result 2H mean in the ontext of the simulation? d Use the simulation results to estimate the proaility that: i Jude wins Srale ii Jude wins adminton iii Jude wins oth games. 2 Two six sided die, one white and the other lak, are rolled onto a tale and the numer uppermost on eah die is reorded. a isplay the sample spae on a lattie diagram. Let e the event of otaining a larger numer on the white die. Find Pr( ). Let e the event that the sum of the uppermost numers on the two die does not exeed 8. What is Pr( )? d Let e the event that oth of the uppermost numers are odd numers. State Pr( ). John Wiley Sons ustralia, Ltd 6

7 e alulate the following. i Pr( Ç Ç ) ii Pr( È ') iii Pr( Ç ) f etermine whether the events and are independent and give a mathematial explanation to justify your answer. 3 Howard an travel to work either y riding his ike or y athing the us. If the weather is foreast as wet, there is a 60 hane that he will drive his ar. However, if the weather is foreast as fine, there is an 80 hane that he will ride his ike. The weather ureau foreasts that the hane of Monday eing wet is 0.4. a onstrut a proaility tree diagram to show the given information and define the symols used. alulate the proaility that Monday is fine and Howard drives his ar to work. (3 marks) John Wiley Sons ustralia, Ltd 7

8 What is the proaility that Howard rides his ike to work on Monday? d Given that Howard rode his ike to work, what is the proaility that Monday was fine? e The weather ureau s reords show that if one day is wet there is a 0.2 hane the next day will e wet; however, if one day is fine, there is a 0.7 hane that the next day will e fine. On Monday, the weather is fine. alulate the proaility that the Wednesday of the same week will also e fine. 4 loal shire ounil has sumissions for 10 new housing developments and 4 new road developments. udget restritions will only allow the ounil to approve 6 of the projet sumissions. a How many different seletions of the 6 projets are possile? alulate the proaility that the 6 projets are for 4 new housing developments and 2 new roads. What is the proaility that at least 2 of the new road developments are approved? (3 marks) d The ounil omes to the deision to approve 4 new housing developments, H1, H2, H3, H, and 2 4 new road developments, R 1 and R. It must then onsider how to shedule the order in whih these 2 developments will e undertaken, one after the other. i In how many different ways an the ounil order the sheduling of these projets? ii What is the proaility that the two road projets R 1 and R are not sheduled with one 2 immediately after the other? (3 marks) John Wiley Sons ustralia, Ltd 8