IB Math High Level Year 1 Probability Practice 1

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1 IB Math High Level Year Probability Practice Probability Practice. A bag contains red balls, blue balls and green balls. A ball is chosen at random from the bag and is not replaced. A second ball is chosen. Find the probability of choosing one green ball and one blue ball in any order. Answer:.. (Total marks). In a bilingual school there is a class of pupils. In this class, of the pupils speak Spanish as their first language and of these pupils are Argentine. The other pupils in the class speak English as their first language and of these pupils are Argentine. A pupil is selected at random from the class and is found to be Argentine. Find the probability that the pupil speaks Spanish as his/her first language. Answer:.. (Total marks). A new blood test has been shown to be effective in the early detection of a disease. The probability that the blood test correctly identifies someone with this disease is 0.99, and the probability that the blood test correctly identifies someone without that disease is 0.9. The incidence of this disease in the general population is A doctor administered the blood test to a patient and the test result indicated that this patient had the disease. What is the probability that the patient has the disease? (Total marks). Given that events A and B are independent with P(A B) 0. and P(A B ) 0., find P(A B). Answer:... (Total marks). A girl walks to school every day. If it is not raining, the probability that she is late is. If it is raining, the probability that she is late is. The probability that it rains on a particular day is. On one particular day the girl is late. Find the probability that it was raining on that day. Answer:.. (Total marks) Macintosh HD:Users:bobalei:Dropbox:Desert:HL:StatProb:Practice:HLProbPractice.docx on /0/0 at : PM Page of

2 IB Math High Level Year Probability Practice. Given that P(X), P(Y X) and P(Y X ), find (a) P(Y ); (b) P(X Y ). Answers: (a)... (b)... (Total marks) 7. In a game, the probability of a player scoring with a shot is. Let X be the number of shots the player takes to score, including the scoring shot. (You can assume that each shot is independent of the others.) (a) Find P(X ). () (b) Find the probability that the player will have at least three misses before scoring twice. (c) Prove that the expected value of X is. (You may use the result ( x) + x + x + x...) () () (Total marks) 8. The probability that a man leaves his umbrella in any shop he visits is. After visiting two shops in succession, he finds he has left his umbrella in one of them. What is the probability that he left his umbrella in the second shop? Answer:... (Total marks) 9. Two women, Ann and Bridget, play a game in which they take it in turns to throw an unbiased sixsided die. The first woman to throw a wins the game. Ann is the first to throw. (a) Find the probability that (i) Bridget wins on her first throw; (ii) Ann wins on her second throw; (iii) Ann wins on her n th throw. (b) (c) (d) Let p be the probability that Ann wins the game. Show that p + p. Find the probability that Bridget wins the game. () Suppose that the game is played six times. Find the probability that Ann wins more games than Bridget. () (Total 7 marks) () () Macintosh HD:Users:bobalei:Dropbox:Desert:HL:StatProb:Practice:HLProbPractice.docx on /0/0 at : PM Page of

3 IB Math High Level Year Probability Practice 0. The probability that it rains during a summer s day in a certain town is 0.. In this town, the probability that the daily maximum temperature exceeds C is 0. when it rains and 0. when it does not rain. Given that the maximum daily temperature exceeded C on a particular summer s day, find the probability that it rained on that day. Answer:... (Total marks). Two children, Alan and Belle, each throw two fair cubical dice simultaneously. The score for each child is the sum of the two numbers shown on their respective dice. (a) (i) Calculate the probability that Alan obtains a score of 9. (ii) Calculate the probability that Alan and Belle both obtain a score of 9. () (b) (i) Calculate the probability that Alan and Belle obtain the same score, (ii) Deduce the probability that Alan s score exceeds Belle s score. () (c) Let X denote the largest number shown on the four dice. (i) x Show that for P(X x), for x,,... (ii) Copy and complete the following probability distribution table. x P(X x) (iii) Calculate E(X). (7) (Total marks). An integer is chosen at random from the first one thousand positive integers. Find the probability that the integer chosen is (a) a multiple of ; (b) a multiple of both and. Answers: (a)... (b)... (Total marks). (a) At a building site the probability, P(A), that all materials arrive on time is 0.8. The Macintosh HD:Users:bobalei:Dropbox:Desert:HL:StatProb:Practice:HLProbPractice.docx on /0/0 at : PM Page of

4 IB Math High Level Year Probability Practice (b) (c) probability, P(B), that the building will be completed on time is 0.0. The probability that the materials arrive on time and that the building is completed on time is 0.. (i) Show that events A and B are not independent. (ii) All the materials arrive on time. Find the probability that the building will not be completed on time. There was a team of ten people working on the building, including three electricians and two plumbers. The architect called a meeting with five of the team, and randomly selected people to attend. Calculate the probability that exactly two electricians and one plumber were called to the meeting. The number of hours a week the people in the team work is normally distributed with a mean of hours. 0% of the team work 8 hours or more a week. Find the probability that both plumbers work more than 0 hours in a given week. (8) (Total marks) () (). The random variable X has a Poisson distribution with mean λ. (a) Given that P(X ) P(X ) + P(X ), find the value of λ. (b) Given that λ., find the value of (i) P(X ); (ii) P(X X ). () () (Total 8 marks). Robert travels to work by train every weekday from Monday to Friday. The probability that he catches the train on Monday is 0.. The probability that he catches the train on any other weekday is 0.7. A weekday is chosen at random. (a) (b) Find the probability that he catches the train on that day. Given that he catches the train on that day, find the probability that the chosen day is Monday. Answers: (a)... (b)... (Total marks) Macintosh HD:Users:bobalei:Dropbox:Desert:HL:StatProb:Practice:HLProbPractice.docx on /0/0 at : PM Page of

5 IB Math High Level Year Probability Practice - MarkScheme. Using a tree diagram, 9 9 R B G. Using a tree diagram, 8 Probability Practice - MarkScheme 8 R B G R B G R B G p(bg or GB) (M)(M) + (A)(A) p(bg or GB) 9 8 (M)(M) (A) (C) English () Argentine () [] Spanish Argentine () () Let p(s) be the probability that the pupil speaks Spanish. Let p(a) be the probability that the pupil is Argentine. (M) Then, from diagram, p(s A) (A) (A) p( S A) p(s A) p( A) (M) (M)(A) (A) Macintosh HD:Users:bobalei:Dropbox:Desert:HL:StatProb:Practice:HLProbPractice.docx on /0/0 at : PM Page of 8

6 IB Math High Level Year Probability Practice - MarkScheme E() A S() 0 (M) p(s A) (A) (A) (C). Let D be the event that the patient has the disease and S be the event that the new blood test shows that the patient has the disease. Let D be the complement of D, ie the patient does not have the disease. Now the given probabilities can be written as p(s D) 0.99, p(d) 0.000, p(s D ) 0.0. (A)(A)(A) [] Since the blood test shows that the patient has the disease, we are required to find p(d S). By Bayes theorem, p( S D) p( D) p(d S) p( S D) p( D) + p( S D ) p( D ) (M) (0.99)(0.000) (0.99)(0.000) + (0.0)( 0.000) (M) ( sf) (A) 0.99 S D D S S 0.9 S (A) Note: Award (A) for 0.99, (A) for 0.000, (A) for 0.0 Therefore p(s) (A) p(d S) (M) ( sf) (A) [] Macintosh HD:Users:bobalei:Dropbox:Desert:HL:StatProb:Practice:HLProbPractice.docx on /0/0 at : PM Page of 8

7 IB Math High Level Year Probability Practice - MarkScheme. Method : (Venn diagram) (M) U A B P(A B) P(A)P(B) P(B) P(B) 0. Therefore, P(A B) 0.8 (M) (A) (C) Method : P(A B ) P(A) P(A B) 0. P(A) 0. P(A) 0. (A) P(A B) P(A)P(B) since A, B are independent P(B) P(B) 0. (A) P(A B) P(A) + P(B) P(A B) (A) (C). Let P(R L) be the probability that it is raining given that the girl is late. L R L L 0 R L P( R L) P(R L) P( L). / P(R L) (M)(A) / + / 0 (using a tree diagram or by calculation) 0 (A) 9 Y X Y X Y Y (a) P(Y ) + (M) [] [] Macintosh HD:Users:bobalei:Dropbox:Desert:HL:StatProb:Practice:HLProbPractice.docx on /0/0 at : PM Page of 8

8 IB Math High Level Year Probability Practice - MarkScheme (A) (C) 0 (b) P(X Y ) P(X Y) (A) (C) 7. (a) P(X ) 9 ( 0. to sf) (M)(A) (b) Let the probability of at least three misses before scoring twice P( m) Let S mean Score and M mean Miss. P( m) [P(0 misses) + P( miss) + P( misses)] (M) [P(SS) + P(SMS or MSS) + P(MMSS or MSMS or SMMS)] (M) + + (A) 89 ( 0.78 to sf) (A) (c) E(x) xp(x) for allx (M)(A) (A) 8. (using the given result) (M) () (A)(AG) First shop Second shop Probability [] [] Left umbrella Did not leave umbrella Left umbrella Did not leave umbrella 9 (M)(A) Required probability 9. (A)(C) (a) (i) P(Bridget wins on her first throw) P(Ann does not throw a ) P(Bridget throws a ) (M) 9 [] Macintosh HD:Users:bobalei:Dropbox:Desert:HL:StatProb:Practice:HLProbPractice.docx on /0/0 at : PM Page of 8

9 IB Math High Level Year Probability Practice - MarkScheme (b) (C) (ii) P(Ann wins on her second throw) P(Ann does not throw a ) P(Bridget does not throw a ) P(Ann throws a ) (M) (C) (iii) P(Ann wins on her nth throw) P(neither Ann nor Bridget win on their first (n ) throws) P(Ann throws a on her nth throw) (M) ( n ). (C) p P(Ann wins) P(Ann wins on her first throw) ++ P(both Ann and Bridget do not win on their first throws) P(Ann wins from then on) (M)(R) + p (C) + p (AG) p P(Ann wins on first throw) + P(Ann wins on second throw) + P(Ann wins on third throw) +. (M) (C) (or ) (C) + p, as required. (AG) (c) From part (b), p p. (C) Therefore, P(Bridget wins) p. (C) (d) P(Ann wins more games than Bridget) P(Ann wins games) + P(Ann wins games) ++ P(Ann wins games) (M) + + (M) ( + + ) 0.. (A) [7] Macintosh HD:Users:bobalei:Dropbox:Desert:HL:StatProb:Practice:HLProbPractice.docx on /0/0 at : PM Page of 8

10 IB Math High Level Year Probability Practice - MarkScheme 0. > (0.) R(0.) R (0.8) > (0.) (M) P(> ) (M)(A) 0.0 P(R > ) (or 0.) 0. 9 (M)(A)(C). (a) (i) P(Alan scores 9) ( 0.) (A) 9 (ii) P(Alan scores 9 and Belle scores 9) 9 8 ( 0.0) (A) (b) (i) P(Same score) (M) 7 ( 0.) 8 (A) (ii) P(A>B) 7 8 (M) 7 ( 0.) 9 (A) (c) (i) x P(One number x) (with some explanation) (R) x P(X x) P(All four numbers x) (M)(AG) (ii) P(X x) P(X x) P(X x l) x x x P(X x) (A)(A)(A) Note: Award (A) if table is not completed but calculation of E(X) in part (iii) is correct. 7 (iii) E(X) (M) (.) (A) 7 [] Macintosh HD:Users:bobalei:Dropbox:Desert:HL:StatProb:Practice:HLProbPractice.docx on /0/0 at : PM Page of 8

11 IB Math High Level Year Probability Practice - MarkScheme. (a) The number of multiples of is 0. (M) Required probability 0.. (A)(C) [] (b) The number of multiples of and is (M) the number of multiples of (A) 8. (A) Required probability 0.08 (A) (C). (a) (i) To be independent P(A B) P(A) P(B) (R) P(A) P(B) (0.8)(0.0) 0. but P(A B) 0. (A) P(A B) P(A) P(B) Hence A and B are not independent. (AG) (ii) [] A B P ( B' A) P(B A) P( A) (M) (M) ( 0.) 7 (A) (b) Probability of electricians and plumber (M) (A) Probability of electricians and plumber!!! (M) ( 0.8) (A) (c) X number of hours worked. X ~ N (, σ ) P(X 8) 0.0 (AG) P(X < 8) 0.90 (M) Φ(z) 0.90 z.8 (z.8) (A) (Answers given to more than significant figures will be accepted.) 0.0 Macintosh HD:Users:bobalei:Dropbox:Desert:HL:StatProb:Practice:HLProbPractice.docx on /0/0 at : PM Page 7 of 8

12 IB Math High Level Year Probability Practice - MarkScheme X µ 8 z >.8 (M) σ σ > σ.9 (Accept σ.8) (A) 0 P(X > 0) P Z > (M).9 0. (A) P(X > 0) 0. (G) Therefore, the probability that one plumber works more than 0 hours per week is 0.. The probability that both plumbers work more than 0 hours per week (0.) (M) 0. (Accept 0. or 0.) (A) 8 λ λ λ. (a) e λ e λ e λ +!!! (M) λ λ 0 λ (A)(A) (b) (i) P(X ) e. e (M)(A) P(X ) 0.89 (G) [] P( X ) (ii) P(X X ) (M) P( X ).. e. e. +..e (A) 0.0 (A) P(X X ) 0.0 (G). (a) Probability (M)(A) 0.7 (A)(C) P(Mon catches train) (b) Probability (M) P(catches train) (A) 0.80 (A) (C) [8] [] Macintosh HD:Users:bobalei:Dropbox:Desert:HL:StatProb:Practice:HLProbPractice.docx on /0/0 at : PM Page 8 of 8

Discrete Random Variable Practice

IB Math High Level Year Discrete Probability Distributions - MarkScheme Discrete Random Variable Practice. A biased die with four faces is used in a game. A player pays 0 counters to roll the die. The