Please solve and show work IB style for 15 and 16. The answer key is attached but the numbering is skewed. I tried to correct the numbering.

36 (1 mark) Tower hears reply b 6 _1 (1 mark) Ship hears 4 6 C (1 mark) 36 Tower does not hear reply 2X_L- (1 mark) 36 18 Ship does not hear e (1,6). (2, 5). (3,4). (4,3). AL- (2 marks) (5,2).(6,1) or using a lattice diagram le conte. . 6 _1 36 6 (2 marks) 3_9 (2 marks) C 1 50 (1 mark) 20 100 2 f since independent 9 90 1- 11 (1 mark) (1 mark) (2 marks) 20 20 100 10 ifi 40 2 8 P(R5 U B5) = (1 mark) P (ship not hear |tower has no reply) = 100 5 P(R5) + P(B5) - P(R5 B5) P (ship not hear ^ tower has no reply) d P(E F) = P(EnF) _10 _1 P (F) 40 4 1 1_1 1 or using a P (tower has no reply 6 6 36 36 (2 marks) lattice diagram (2 marks) S (2 marks) e If independent then h Considering the list in e P(An B) = 0 so events P(E|F) = P(E) (1 mark) e 2 = - or using conditional are mutually exclusive 6 3 (2 marks) 1 60 4 - so not independent probability formula 30x2 = 18 100 (2 marks) (1 mark) (2 marks) Independent 6 50x= = 20 (2 marks) P(FOR) = P(F) X P(R) catch 20% (1 mark) C TX= = 30 = T=50 (2 marks) injected 30% 1 1x - so not 6 3 4 38 . Let x be the number not catch 80% independent 1 mark) speaking both English and French. catch 90% P(F UR) = P(F) + P(R) - P(F ~ R) (60-x)+x+(40-x)+10 = 100 not Injected 70% (1 mark) =110-x = 100 = x =10 1 1 1_5 (2 marks) 3 4 6 12 (1 mark) not catch 10% (4 marks) (2 marks layout P(exactly one team) = 2 marks numbers) 5 1_ 1 (2 marks) 12 6 4 70 90 63 50 10 30 10 Could also use a Venn 100 100 100 (1 mark) diagram in (b) and (c). 30 80 6 d P ( FOR| F ) = - P(FOR) 6=1 100 100 25 (1 mark) P(F) -2 A3 30 20 70 90 69 (2 marks) (1 mark shape 2 marks 100 100 100 100 100 14 48 a numbers) (2 marks) 689ANSWERS IT ant. .. . Is. cont. 122 61 (1 mark) Exercise 94 P(PC) - P(7'nC)_ _ 2 200 100 P(C) 23 92 23 (1 mark) (2 marks) 200 50 3 2able 4 21712 82 118 59 6 27r's" P(IC) = -P(InC) 1 - 24 200 200 100 2 marks) P(C') 8 x14y 20 3 or by using the 12 marks) 48 12 formula (2 marks) 9 12 1 9x4y 12 1 0a* 22 42 marks 14 Exercise 98 8 RG or GR 15 * V7 b /2' V6' 28 12 marks (60) 5 5 25 ( 3x) 8 8 65 42 marks V.x' (4 marks) (Al shape. 3 marks. I RG or GR 3, 3 3 5 15 42 3 7 numbers. 42 marks 32 2 a 10- 4 numbers, I mark 2 numbers) 12 marks) (5x ) 1 D 200-140-60 12 marks) . (2d) 4 f 3.x3 3 x 2 30 200 20 (1 mark) Exercise 9C 2 20 8 1 P(A B) AND- PLANB) P(B) 0.5 2 " P(AnB) = 0.2 (2 marks) " P(A) = P(An B)+ P(An8) = 0.2+0.4 =0.6 (2 marks) 7 "P(AVB) = P(A)+P(8)-P(A~.B)= 0.6+0.5-0.2=0.9 (2 marks) Exercise 90 NV P(A B)= P(AnB') 0.4 a Growth b Decay 2 = 0.8 P[B') 0.5 (2 marks) c Decay d Decay . Growth b PAJA) . P(A|B' ) so not independent (2 marks) 2 (x) = D g(x) = C hix) = A its = B jux) = E . . . . . . . . ....... ................. ..... Chapter 9 Skills check 1 a 32 b 1000 125 d 243 216 IN y'=4 V X E R. y > 4 6908 QUANTIFYING RANDOMNESS: PROBABILITY g one of the dice is a 5 (2 marks) 15 P1: On average in Wales it rains on 3 days h one of the dice is a 5 given that the out of every 5 days. total of the two dice is 7. (2 marks) a Over a month, that has 30 days in 13 P1: The probability that James is selected it, calculate how many of these days would be expected to have rain. for the school football team is 1. The 3 (2 marks) probability that he is selected for the Over a period of 50 days, calculate how many days would not be school rugby team is -. The probability 4 expected to have rain. (2 marks) that he is selected for the both teams is _. c Calculate what length of time in a State, with a reason, whether the 6 days, would be expected to have 30 events of him being selected for days of rain in it. (2 marks) football and being selected for rugby 16 P1: At an international school with 100 are independent. (2 marks) students, 60 students speak English, 40 b Find the probability that he is selected speak French and 10 students speak for at least one team. (2 marks) neither of these two languages. c Find the probability that he is selected a Calculate how many students speak for exactly one team. (2 marks) both English and French. (2 marks) d Find the probability that he is b Sketch a Venn diagram to represent selected for both teams given that he this information. In each enclosed is selected for football. (2 marks) region put in the number of students corresponding to this region. 14 P1: A control tower sends a message to a (3 marks) ship. The probability that the ship hears Calculate the probability that a the message is -. The ship will reply student chosen at random if and only if it receives the message. If i speaks English but not French the ship replies, the probability that the ii speaks either English or French tower hears the reply is 3 iii does not speak English. 5 a Sketch a probability tree to represent (3 marks) this information. (2 marks) d Given that a student speaks French, b Hence find the probability that the find the probability that they also speak tower hears a reply to the message it English. (2 marks) sends. (2 marks) e Let E be the event that a student c Write down the probability that it speaks English and F be the event that does not hear a reply. (2 marks) a student speaks French. Determine with a reason if events E and F are d Given that the tower did not hear a independent or not. (2 marks) reply, find the probability that the ship did not hear the original message. 17 P2: A disease that spreads rapidly is about (2 marks) to arrive at a city with a very large number of inhabitants. If a person does e Events are defined as follows: not do anything, then the probability A: The ship does not hear the original that they will catch the disease is 90%. message. If they have taken a special injection B: The tower receives a reply. then the probability that they will catch State, with a reason, whether the two the disease reduces to 20%. Sadly only events A and B are mutually exclusive 30% of the population could afford the or not. (2 marks) injection and they were all injected. 3908 a Sketch a probability tree that 19 P2: Adrian has a box with 5 red and 3 green represents all this information. apples in it. He takes out an apple at (4 marks) random and eats' it. He then takes out Find the probability that a random another apple at random and also eats it. person is not injected and catches the a Find the probability that he eats disease. (1 mark) i two red apples Find the probability that a random ii two apples of different colours. person is injected and does not catch the disease. (1 mark) (4 marks) d Find the probability that a random Sally also has a box with 5 red and 3 person does catch the disease. green apples in it. Sally is on a diet. She (2 marks) takes out an apple at random and then puts it back in the box. She then takes e Given that a random person out another apple at random. has caught the disease find the probability that they were not b Find the probability that she takes injected. (2 marks) i two red apples f Given that a random person has ii two apples of different colours. not caught the disease find the (4 marks) probability that they were injected. (2 marks) 20 P1: a For two events, A and B, 18 P1: A college with 200 students offers P( B) = 0.5, P( A|B) = 0.4 and rowing, kayaking and surfing as P(An B') = 0.4. activities. Of the students, 8 do all three Calculate activities, 30 do rowing and kayaking, 20 do rowing and surfing, 10 do surfing i P(An B) and kayaking but not rowing, 40 do ii P( A) only rowing, 70 do kayaking and 48 do surfing. ili P( A U B) a Sketch a Venn diagram to represent iv P( A B' ) (8 marks) Statistics and probability all this information. In each enclosed b Determine, with a reason, whether space in the diagram put the number events A and B are independent or of students that this space specifically not. refers to. (4 marks) (2 marks) Calculate how many students do not do any of these three activities. (2 marks) c If a student is chosen at random find the probability they i do only kayaking ii do rowing or kayaking iii do rowing or kayaking but not surfing iv do not do rowing. (5 marks) d Given that a student does surfing find the probability that they also do rowing. (2 marks) 391