For use only in Whitgift School. IGCSE Higher Sheets 8. IGCSE Higher

Transcription

1 IGCSE Higher Sheet H Transformations Sheet H Transformations Sheet H Transformations Sheet H a- Histograms Sheet H a- Histograms Sheet H a- Histograms Sheet H a-4 Histograms Sheet H a- Cumulative Frequency Sheet H a- Cumulative Frequency Sheet H a- Cumulative Frequency Sheet H8-6-0a-4 Cumulative Frequency Sheet H8-6-0a- Tree Diagrams Sheet H8-6-0a- Tree Diagrams Sheet H a- Tree Diagrams Sheet H b- Probability-Independent Events Sheet H b- Probability-Independent Events Sheet H c- Conditional Probability Sheet H c- Conditional Probability Sheet H c- Conditional Probability Sheet H c-4 Conditional Probability

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3 Sheet H Transformations. On a set of axes with x ranging from -6 to 8 and y ranging from 6 to 0 (cm per unit), draw and label the triangle T with vertices at A( 0, ), B(, ) and C(, ). On the same set of axes draw and label the following: (a) The triangle T with vertices A, B and C which is obtained by translating T by the 4 vector. (b) The triangle T with vertices A, B and C which is obtained by an enlargement of T by scale factor, the centre of enlargement being (, ). (c) The triangle T with vertices A, B and C which is obtained by reflecting T in the line y = x +. (d) The triangle T 4 with vertices A 4, B4 and C4 which is obtained by rotating T about the point (, ) by 90 in the clockwise direction.. On a set of axes with x ranging from -6 to 0 and y ranging from 8 to 8 (cm per unit), draw and label the triangle T with vertices at A (, 6), B(7, 6) and C(6, 7). (a) On the same set of axes draw and label: (i) the triangle T with vertices A, B and C which is obtained by enlarging T by scale factor and centre of enlargement (5, 4). (ii) On the same set of axes draw and label T with vertices A, B and C which is obtained by rotating T about the point (, ) through 90 in the anticlockwise direction. (iii) The triangle T with vertices A (,0), B (, 4) and C( 4, ). (iv) The triangle T 4 with vertices A 4, B4 and C4 which is obtained by rotating T about the point (-6, -5) through 90 in the clockwise direction. (v) The triangle T 5 with vertices A 5(, 8), B5 (7, 8) and C5(6, 7). (b) (i) Find the equation of the line through which T is reflected onto T. (ii) Find the equation of the line through which T 4 is reflected onto T 5. (iii) Find the vector through which T is mapped to T 5.

4 Sheet H Transformations. On a set of axes with x and y ranging from -8 to 8 (cm per unit), draw and label the following triangles: T with vertices at A (, 0), B(, ) and C(6, ) T with vertices at A (, ), B (, 4) and C (, 7) T with vertices at A ( 8, 6), B (6, 5) and C (5, ) T with vertices at A, 6), B (, ) and (5, 0) ( C (a) Find the equation of the line of reflection about which T is reflected onto T. (b) Find the centre of rotation about which T is rotated onto T and find the angle of rotation. (c) Find the centre of enlargement about which T is enlarged onto T and find the scale factor of enlargement.. On a set of axes with x and y ranging from -8 to 8 (cm per unit), draw and label the following triangles: T with vertices at A (, ), B(8, ) and C(4, 0) T with vertices at A ( 4, ), B ( 4, 7) and C (, ) T with vertices at A 6, ), B (6, 7) and C (4, ) ( It is given that T is mapped onto T by a rotation about the origin followed by a translation. (a) Find the angle of rotation. (b) Find the column vector of translation. (c) Find the centre of rotation about which T is rotated onto T and find the angle of rotation. (d) Given that T is mapped onto T by a rotation of 90 anticlockwise about (, ) followed by a translation, find the column vector of translation. (e) Draw and label the triangle T which is obtained by an enlargement of T, with centre of enlargement (, ) and scale factor.

5 Sheet H Transformations. Draw a set of axes with x and y both going from -8 to 8 with cm per unit on both axes. (a) Draw the rectangle R with vertices A(, ), B(6, ), C(6, 4) and D (, 4). (b) Draw the rectangle R with vertices A (-4, 4), B (-4, 0), C (-, 0) and D (-, 4). (c) Find the angle and centre of the rotation which maps R onto R. (d) Draw the rectangle R with vertices A, B, C and D which is a reflection of R in the line y = x. (e) Draw the rectangle R with vertices A (8, -4), B (0, -4), C (0, -8) and D (8, -8). (f) Find the scale factor and centre of the enlargement which maps R onto R. (g) Draw the rectangle R 4 with vertices A 4, B 4, C 4 and D 4 which is obtained by rotating R through an angle of 90 in the anticlockwise direction about the point (-5, 5). (h) What is the equation of the line through which R 4 is mapped to R?. Draw a set of axes with x and y both the x from -8 to 8 and the y from 8 to (with cm per unit on both axes). (a) Draw the triangle T with vertices A(5, ), B(7, 4) and C(6, 6). (b) The triangle T with vertices A, B and C which is an enlargement of T with scale factor and centre of enlargement (, 5). (c) Draw T with vertices A, B and C which is an enlargement of T with scale factor and centre of enlargement (4, ). (d) Find the centre and scale factor of the enlargement which maps T to T. (e) Draw T with vertices A, B and C which is obtained by rotating T through 90 in the clockwise direction about the point (, 0). (f) Draw T 4 with vertices A 4(, 6), B4 (, 8) and C4(, 7). (g) Find the angle and centre of the rotation which maps T onto T 4. (h) Find the vector which translates T 4 onto T.

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7 Sheet H a- Histograms. (a) Copy and complete the following table which shows the time taken by a group of candidates to finish an exam: Time in minutes 45 x < x < x < x < x < 90 Number of candidates Frequency Density 4. 5 = (b) Draw a histogram to illustrate this data, using a scale of cm per 5 minutes on the horizontal axis (TIME - which goes from 45 to 90) and cm per unit on the vertical axis (FREQUENCY DENSITY - which goes from 0 to 8). Label your axes clearly.. The height in cm of plants is shown below: Height (cm) Frequency Class Width Frequency Density 6. 5 = = (a) (b) Copy and complete the above table. Draw a histogram to illustrate this data, using a scale of cm per 5 units on the horizontal axis (which goes from 5cm to 60cm) and 5cm per unit on the vertical axis (which goes from 0 to.5). PTO

8 Sheet H a- Histograms (cont.). The table below shows the ages of the people stopped by a company investigating the voting intentions of the adults in a certain town: Age Age (inequality) 8 a < 50 a < 75 Frequency Frequency Density.5 (a) Explain why the 8- category is represented by the inequality 8 a <. (b) Copy and complete the table shown above. (c) Draw a histogram to illustrate this data, using a scale of cm per 5 units on the horizontal axis (which goes from 5 to 75) and cm per unit on the vertical axis (which goes from 0 to 6).

9 Sheet H a- Histograms. At a girls school, a random sample of 0 pupils was taken and each pupil recorded her intake of milk (in ml) during a given day. The results are shown below: Milk intake (ml) No. of students (a) (b) Calculate the frequency density for each of the above classes. Draw a histogram to illustrate these data using cm per 50ml on the horizontal axis and cm per 0. on the vertical axis.. Summarised below are the prices of the goods (to the nearest ) sold by an electrical shop on a certain day. Price of goods ( ) Frequency Frequency Density (a) Explain why the bar on the histogram for 0-9 should go from 9.5 to 9.5 (b) Copy and complete the above table. (c) Draw a histogram to illustrate these data using cm per 0 on the horizontal axis (from 9.5 to 9.5) and cm per unit on the vertical axis (from 0 to 7).. A boy drew a histogram for the data shown below. He included columns for the height and the width of each bar of his histogram.. Weight (kg) Frequency Frequency Density Height on Histogram (cm) Width on Histogram (cm) cm cm On the boy s histogram the 60- bar was cm wide and 8cm tall as shown in the above table. (a) Copy and complete the above table. (b) Hence draw the boy s histogram using the scale that he used.

10 Sheet H a- Histograms. Measurements of the time intervals between successive arrivals of telephone calls at an office exchange were taken. The first 00 time intervals were recorded and the following grouped frequency distribution was obtained. Time interval Frequency (x mins) 0 x < x <.0.0 x <.0.0 x < x < (a) (b) Set up a frequency density table. Draw a histogram to illustrate this distribution, using a scale of cm per minute on the horizontal and cm per 5 units on the vertical scale [JMB, U&C]. The table given below shows a grouped frequency distribution of the recorded heights, measured to the nearest centimetre of 50 girls. Height (cm) No. of girls (a) Explain why the 0-05 class when written as an inequality is 05.5 h < 05.5 where h is the height. (b) Explain also why the frequency density for the 0-05 class is.5. (c) Set up a frequency density table for the above data. (d) Draw a histogram to illustrate these data, using a scale of cm per unit on both the horizontal and the vertical scale. [WJEC, U&C]. The table below shows the age (at last birthday) at which women married in 986 in England and Wales.. Age (in yrs) Women (in tens of thousands) 6 8 Frequency Density (a) Explain why the 6-0 class is written as 6 A < where A is age. (b) Explain also why the frequency density for the 6-0 class is.. (c) Copy and complete the above table. (d) Draw a histogram to illustrate these data, using a scale of cm per year on the horizontal and cm per unit on the vertical scale. [O&C, U&C]

11 Sheet H a-4 Histograms. The table below shows the time taken by a group of candidates to finish an exam. Time in minutes Number of candidates Frequency Density 4. (a) Explain why the class when written as an inequality is 44.5 t < 49.5 where t is the time. (b) Explain also why the frequency density for the class is 4.. (c) Copy and complete the above table. (d) Draw a histogram to illustrate this data, using a scale of cm per 5 units on the horizontal axis (from 44.5 to 79.5) and cm per unit on the vertical axis.. The height of a certain type of flower was measured (to the nearest cm) and the results are shown below: Height (cm) Frequency Frequency Density 6.8 (a) Explain why the 6-0 class when written as an inequality is 5.5 h < 0.5 where h is the height. (b) Explain also why the frequency density for the 6-0 class is 6.8. (c) Copy and complete the above table. (d) Draw a histogram to illustrate this data, using a scale of cm per 5 units on the horizontal axis (from 5.5 to 60.5) and cm per unit on the vertical axis.. The table below shows the ages of the people stopped by a company investigating the voting intentions of the adults in a certain town: Age Frequency Frequency Density (a) Explain why the 8- class when written as an inequality is 8 A < where A is the age. (b) Explain also why the frequency density for the 8- class is.5. (c) Copy and complete the table shown above. (d) Draw a histogram to illustrate this data, using a scale of cm per 5 units on the horizontal axis (from 0 to 80) and cm per unit on the vertical axis.

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13 Sheet H a- Cumulative Frequency. The table below shows the mass of 40 eggs laid at a certain farm: Mass (g) m < m < m < m < 6 6 m < m < 74 Frequency (a) Copy and complete the following table: Mass (g) Cumulative Frequency 0 50 (b) (c) (d) Draw a cumulative frequency graph for this data using a scale of cm per 5 units on both axes. (The horizontal axis should go from 0g to 80g). Use the graph to find an estimate of the median. Use the graph to find estimates of the lower and upper quartiles.. The length of each of 50 electric light bulbs is noted and the results shown in the table below. Length (h) Frequency (a) Copy and complete the following table: Length (h) Cumulative Frequency (b) Draw a cumulative frequency graph using a scale of cm per 5 units on the horizontal axis and cm per 5 units on the vertical axis. (The horizontal axis should go from 650h to 70h). (c) Use the graph to find an estimate for the median and the interquartile range. PTO

14 Sheet H a- Cumulative Frequency (cont.). The table below shows the time it took a group of 0 pupils to finish their Maths exam: Time taken (min) 55 t < t < t < t < 75 Number of boys (a) (b) (c) (d) Complete a cumulative frequency table. Draw a cumulative frequency graph using a scale of cm per 5 units on the horizontal axis and cm per 0 units on the vertical axis. (The horizontal axis should go from 55min to 75min). How many boys finished the exam in under 6 minutes? How long did it take the first 90 pupils to finish the exam? 4. The weights (in kg) of 50 players in a rugby competition were recorded in the able below: Weight (kg) Frequency (a) (b) (c) (d) Complete a cumulative frequency table. Draw a cumulative frequency graph using a scale of cm per 5 units on the horizontal axis and cm per 0 units on the vertical axis. (The horizontal axis should go from 70kg to 0kg). Use the graph to find an estimate for the interquartile range. From your graph find the probability that a player weighs more than 5kg.

15 Sheet H a- Cumulative Frequency. The masses (in kg) of 00 animals were recorded in the table below: Mass (kg) Frequency Write down the coordinates of the points through the cumulative frequency curve should pass.. The mass of a group of 80 competitors in a sports tournament are recorded below: Mass (kg) Frequency (a) (b) (c) (d) (e) Without any calculations, state in which class width the median must lie. How many competitors had a mass below 55kg? How many competitors had a mass below 85kg? Draw a cumulative frequency graph. Use a scale of cm per 5 units on the horizontal axis and cm per 5 units on the vertical axis. (The horizontal axis should go from 55kg to 90kg). Use the graph to find an estimate for the median.. A biologist recorded the heights of a group of 00 insects and displayed his results in the table shown below: Height (mm) Frequency (e) (f) (g) Draw a cumulative frequency graph for this data using a scale of cm per mm on the horizontal axis and cm per 0 units on the vertical axis. (The horizontal axis should go from 0mm to 7mm). Use the graph to find an estimate of the median. Use the graph to find estimates of the lower and upper quartiles. 4. The table below records the time taken for a group of 0 candidates to finish a task: Time taken (min) 5 t < 0 0 t < 5 5 t < 0 0 t < 5 5 t < 0 Number of candidates (e) (f) (g) (h) Draw a cumulative frequency graph using a scale of cm per 5 units on the horizontal axis and cm per 0 units on the vertical axis. (The horizontal axis should go from 5min to 0min). Use the graph to estimate the median time taken in the above data. Estimate how long it took for the first 50 pupils to finish the exam. Estimate how many candidates finished in under 8 minutes.

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17 Sheet H a- Cumulative Frequency. The lengths (in cm) of 50 metal poles were recorded in the table below: Length (cm) Frequency (a) Copy the following table, add in the missing rows and complete it. Length (cm) Cumulative Frequency < 0 < (b) (c) Draw a cumulative frequency graph using a scale of cm per unit on the horizontal axis and cm per 5 units on the vertical axis. Use the graph to find an estimate for the inter-quartile range.. The speeds, in km/h, of 50 cars recorded on the M was as following: Speed (km/h) Frequency (a) (b) (c) (d) Copy the following table, add in the missing rows and complete it. Speed (km/h Cumulative Frequency Not exceeding 85 Not exceeding 95 Draw a cumulative frequency graph for this data using a scale of cm per 5 units on the horizontal axis and cm per 0 units on the vertical axis. Use the graph to find an estimate of the median speed. Estimate the probability of a car on the M travelling between 90 km/h and 0 km/h. PTO

18 Sheet H a- Cumulative Frequency (cont.). A survey of 00 student houses was made by a university and the prices of the houses were recorded: Price ( ) Cumulative Frequency Up to 0,000 0 Up to 40,000 8 Up to 50, Up to 60,000 Up to 70,000 6 Up to 80,000 8 Up to 90,000 9 Up to 00, (i) (j) (k) (l) The cumulative frequency curve must start on the horizontal axis. What are the coordinates of the starting point of the curve? Draw a cumulative frequency graph using a scale of cm per 0,000 on the horizontal axis and cm per 0 units on the vertical axis. Use the graph to estimate the inter-quartile range. Estimate the price below which 60% of houses fall.

19 Sheet H8-6-0a-4 Cumulative Frequency. The lengths of 5 ears of barley (to the nearest mm) gave the following frequency distribution: Length (mm) Frequency (a) What is the upper bound for the 5-9 interval (BE CAREFUL)? (b) Draw a cumulative frequency table and graph for this data using a scale of cm per 5 units on the horizontal axis and cm per 0 units on the vertical axis. (c) Use the graph to find the number of ears of barley which were less than 57mm long (show your working on the graph). (d) Use the graph to find the length which 75 ears of barley are below long (show your working on the graph).. The table below shows the number of runs scored by a group of 5 boys in a season of school cricket matches. Runs Frequency (a) Copy and complete the following cumulative frequency table: Runs Cumulative Frequency (b) (c) (d) Draw a cumulative frequency graph for this data using a scale of cm per 0 units on the horizontal axis and cm per 0 units on the vertical axis. Obtain an estimate of the median from your graph. From your graph, estimate the interquartile range (show your working on the graph).

20 Sheet H8-6-0a- Tree Diagrams. The probability of a pupil from a school passing an exam is 0.4. Two pupils are chosen at random. (a) Draw a tree diagram to show all the possibilities. (b) Find the probability that: (i) Both pupils pass the exam. (ii) Only one passes. (iii) At least one passes the exam.. A bag contains 7 balls. are red and the rest are green. Two balls are chosen at random and the first is replaced before the second is chosen. (a) Draw a tree diagram to represent this information. (b) Find the probability that: (i) the two balls are red. (ii) the two balls are the same colour. (iii) exactly one of the balls is green.. Each day a man has to ring an information phone line on two occasions. The probability of the line being engaged is ¼. (a) Display this information on a tree diagram. (b) Use this diagram to find the probability that : (i) On both occasions the line is engaged (ii) the line is not engaged on at least one of his calls. 4. At a fairground a man offers a small prize if the contestant rolls a six on a fair die. A man has two goes at the game. (a) Display this information on a tree diagram, distinguishing between a prize on one branch and not a prize on the other branch. (b) What is the probability that the man wins two prizes? (c) What is the probability that the man wins at least one prize? (d) What is the probability that the man wins no prizes at all? 5. The probability of a new born baby being a girl is 0.5. A family has two children. (a) Draw a tree diagram to show all the possibilities. (b) Find the probability that: (i) Both children are girls. (ii) There is at least one boy. (iii) Both children are the same sex. 6. The probability of being left-handed is 0.. (a) Draw the tree diagram to show all the possibilities when three people are chosen at random. (b) Find the probability that: (i) There is exactly one left-handed person. (ii) There are at least two right-handed people. 7. The probability of a man being late to a meeting is 0%. One day he has two meetings (a) Display this information on a tree diagram. (b) Use this diagram to find the probability that: (i) he was late to both meetings. (ii) he was late to neither meeting.

21 Sheet H8-6-0a- Tree Diagrams. The probability that Susannah is late for school is 0.4; the probability that Claire is late is 0.7. (a) What assumption do you need to make to calculate the probability that both girls are late for school? (b) Making this assumption, draw a tree diagram to show all the possibilities. (c) Calculate the probability that both girls are late for school. (d) Calculate also the probability that only one of the two girls is late for school.. In a car factory, 5% of the cars have a minor fault with the locking and 0% have a minor problem with the windscreen wipers. The two faults are independent of each other. (a) Draw a tree diagram to show all the possibilities. (b) One car is chosen at random from this factory. Find the probability that this car: (i) did not have a fault with either the locking device or the windscreen wipers. (ii) had one fault with either the locking device or the windscreen wipers.. The probability that the local Grammar School beats the comprehensive at the annual rugby match is 0.7 and at football is 0.. These events are independent. (a) Draw a tree diagram to show all the possibilities (do not consider draws). (b) Calculate, the probability that, in a certain year: (i) the grammar schools wins both matches. (ii) the grammar schools wins at least one of the matches. (c) Over three years the Grammar schools fails to win any of the matches. 4. Once a month Anthony checks the water and oil in his car. The probability that the oil needs topping up is 0., the probability that the water needs topping up is 0.5. (a) Draw a tree diagram to show all the possibilities. (b) Find the probability that, the next time Anthony checks his car, (i) Both the oil and the water need topping up. (ii) The oil needs topping up but not the water. (ci) Neither the oil nor the water needs topping up. (cii) At least one of the oil or water needs topping up. 5. In the Premier League, a team scores points if it wins, point if it draws, 0 points if it loses. The probability that Liverpool will win their next match is 0.6. The probability that it will lose its next match is 0.. Find the probability that: (a) (b) Liverpool will gain at least point from their next game. Liverpool will gain at least 4 points from their next two games, assuming that the probabilities associated with their second game are unchanged.

22 Sheet H a- Tree Diagrams. The probability of a set of traffic lights being red is ¾. A bus passes through these lights twice on its journey, (a) Draw a tree diagram to show all the possibilities (do not consider amber). (b) Find the probability that: (iv) The lights were red on both occasions (v) On neither occasion the lights were red.. A bag contains 0 balls - are black, are red and the rest are green. Two balls are chosen at random and the first is replaced before the second is chosen. (a) Draw a tree diagram to represent this information. (b) Find the probability that: (i) the two balls are green. (ii) neither of the two balls is red. (iii) the balls are both the same colour.. Each day a man has to pass through the same set of traffic lights three times. He reckons that the probability of the lights being red is 0.4. (c) Display this information on a tree diagram. (d) Use this diagram to find the probability that : (j) all three sets show red (iii) only one set shows red. (iv) at least one of them is red. 4. A bag contains balls - 4 are red, 6 are blue and the rest are yellow. Two balls are chosen at random and the first is replaced before the second is chosen. (a) Draw a tree diagram to represent this information. (b) Find the probability that: (i) the two balls are different colours. (ii) neither of the two balls is red. (iii) the balls are both the same colour.

23 Sheet H b- Probability-Independent Events. Two coins are tossed. Copy and complete the following table Coin Coin H T H HH T Find the probability of : (a) getting two heads (b) getting two tails (c) getting a head and a tail (in either order) (d) not getting two tails.. A coin is tossed and a die is thrown. Find the probability of getting: (a) a multiple of on the die and a head on the coin. (b) a tail on the coin and an odd number on the die.. Two dice are thrown. Their scores are added together. Copy and complete the following table: (a) Find the probability of getting two sixes (b) Find the probability of getting one five and one four (in either order) (c) Find the probability of getting a total score of 0 (d) Find the probability of getting the same number on both dice. (e) What is the most likely score? What is probability of getting this score? 4. A box contains counters. are blue, are red and the rest are purple. Two counters are chosen at random, the first being replaced before the second is chosen. Find the probability that (a) the first counter is purple and the second is blue (b) both counters are red (c) one of the counters is blue and the other is red. PTO

24 Sheet H b- Probability-Independent Events (cont.) 5. Three coins are tossed at the same time. (a) List all the eight possible outcomes, using H and T. (b) Find the probability of: (i) getting three tails (ii) getting the same outcome on all three coins (iii) getting two heads and one tail (in any order) (iv) UnotU getting three heads 6. A coin is biased so that heads appear twice as often as tails. (a) If p is the probability of getting a tail, write down the probability of getting a head in terms of p. (b) Hence show that p =. (c) The coin is tossed twice. Find the probability of getting: (i) two heads (ii) a head first and then a tail (iii) two tails 7. (a) Without listing them, write down how many possible outcomes there are when six coins are tossed together? (b) What is the probability of tossing six heads?

25 Sheet H b- Probability-Independent Events. Mark works in a computer sales office which sells British, Japanese and American computers. 45% of the computers that Mark sells are British and 0% are Japanese. (a) What is the probability that Mark sells an American computer to his next customer. (b) Find the probability that the next two customers both buy British computers. (Assume these customers do not know each other.) (c) Why do you need to make the assumption in (b)? (d) What is the probability that at least one of Mark s next four customers (none of whom know each other) buy a British computer. (HINT: Consider the only other possibility).. (a) Draw a 6 by 6 grid to show the 6 equally likely total outcomes when two fair dice are rolled. (b) How many outcomes give a total of 5? (c) Find the probability of getting a total of: (i) 7 (ii) 4 (iii) (d) (e) What is the probability of getting the same number on both dice. Which number has the greatest probability associated with it and what is the probability of getting this number?. Three ordinary dice are rolled. Find the probability that: (a) All three dice show the same number (b) The total score is exactly 8 (c) The total score is exactly (a) In a board game the number of squares a player moves is equal to the total shown when two dice are thrown. (b) Mark is 8 squares away from landing in Jail. What is the probability that he avoids this within his next throw? (c) Mark is squares away from landing on one of James property, away from landing on another of his properties and 5 from James third property. What is the probability that he lands on one of these properties on his next throw? (d) James is fewer than 6 squares away from landing on the nearest of Mark s properties. There is one square between Mark s second property and his first one. Find how many squares James is from Mark s nearest property given that the probability of James landing on either of Mark s property is 5/8. PTO

26 Sheet H b- Probability-Independent Events (cont.) 5. A machine drops dots of paint at random onto a circle of radius 0cm. In the middle of this circle is a black circle of radius 5cm. (a) Show that the probability that the drop of paints falls within the black circle is ¼. (HINT: Use calculations in terms of π ). (b) Find, as a fraction, the probability that when two drops of paint fall, only one of them falls into the black circle. (c) Find, as a fraction, the probability that when two drops of paint fall, at least one of them falls into the black circle. (d) Find, as a fraction, the probability that when three drops of paint fall, none of them falls into the black circle. 6. A card is drawn at random from a pack of 5 playing cards. It is then returned and a second card is chosen. Find the probability that a king and an ace are chosen (in any order).

27 Sheet H c- Conditional Probability. A bag contains 9 balls. 4 are red and the rest are blue. A ball is removed at random and then another ball is removed. (a) Set up a tree diagram showing the probabilities of all the possible outcomes. (b) Find the probability that (i) both balls are blue. (ii) both balls are the same colour (iii) at least one of the balls is red.. A bag contains coins. Five are 50p coins and the rest are 0p coins. A coin is removed at random and then another coin is removed. (a) Set up a tree diagram showing the probabilities of all the possible outcomes. (b) Find the probability that (i) the total of the coins is exactly 70p (ii) the total of the coins is less than (iii) the total of the coins is more than 40p. A bag contains 0 balls. Five are red, three are blue and the rest are green. A ball is removed at random and then another ball is removed. (a) Set up a tree diagram showing the probabilities of all the possible outcomes. (b) Find the probability that (i) both balls are red (ii) both balls are the same colour (iii) at least one of the balls is blue % of the population of a certain town are vaccinated against flu. The probability of someone getting the flu given that they have had the vaccination is 0. but the probability of someone getting flu given that they have not had the vaccination is 0.7. (a) Draw a tree diagram to represent the above information. (b) (c) Find the probability that a person chosen at random gets flu. Write down the probability that a person chosen at random doesn t get flu given that he didn t have the vaccination. 5. blue balls and 5 red balls are put into a bag and two balls are selected, one at a time, without replacement. (a) Draw a tree diagram to represent the above information. (b) Find the probability that both balls are the same colour. (c) Find the probability that the second ball is blue given that the first is red. 6. A hockey team plays 45% of its matches at home. It wins 0% of its away matches and 60% of its home matches. (a) Draw a tree diagram to represent the above information. (b) Find the probability that a match chosen randomly from the results sheet at the end of the season was won. (c) Find the probability that a match chosen randomly from the results sheet at the end of the season was an away match which was not won. PTO

28 Sheet H c- Conditional Probability (cont.) 7. 0% of items in an antique shop are fakes. The probability of an expert declaring an item to be fake given that it is a genuine is 0. and the probability of him declaring an item to be genuine given that it is a fake is (a) Draw a tree diagram to represent the above information. (b) Find the probability that an item chosen at random will be a fake but will be declared genuine. (c) Find the probability that an item will be declared genuine.

29 Sheet H c- Conditional Probability. A football team plays two matches per week. The probability that it wins the first match is ¾. If it wins the first match then the probability of winning the second match is 4 5, but if it does not win the first match then the probability of winning the second match is it is 5. (d) Draw a tree diagram to represent the above information (with second match after the first match). (e) Find the probability that the team wins both matches. (f) Read off from your tree diagram the probability that the team doesn t win the second match given that it didn t win the first match.. The probability of there being road works on a man s way to work is 5%. If there are road works then the probability of a man being late to work is 80%, otherwise it is 40%. (a) Draw a tree diagram to represent the above information (with road works first). (b) Find the probability that, on a day chosen at random, there were road works and he was on time. (c) Find the probability that, on a day chosen at random, he was late.. A football team plays 45% of it matches at home. The probability of winning a match given that it is at home is 0.7. The probability of winning a match given that it is at away is 0.6 (a) Draw a tree diagram to represent the above information. (b) Find the probability that the team lost a match. 4. The probability of a speed camera being on is If the camera is on the probability that a speeding driver will be caught is 0.8. If the camera is off then the probability of a speeding driver being caught is 0. (a) Draw a tree diagram to represent the above information. (b) Find the probability that a speeding driver does not get caught. 5. Three red balls and four blue balls are placed in a bag and two balls are removed one at a time and are not replaced. (a) Draw a tree diagram to show the probability of every possible outcome. (b) Find the probability that the two balls were different colours.

30 Sheet H c- Conditional Probability. Five yellow balls and three red balls are placed in a bag and two are removed, one at a time, without replacement. (a) Draw a tree diagram to represent the above information. (b) Find the probability that: (i) both balls are red. (ii) both balls are the same colour. (iii) the second ball is yellow given that the first is red.. A boy is late 60% of the time when it is raining and 0% of the time when it is dry. It rains on 5% of days. (a) Draw a tree diagram to represent the above information. (b) Find the probability that: (i) It is raining and he is late. (ii) He is late. (iii) He is on time given that it is dry.. A team plays 40% of its matches at home. They win 75% of their home matches but only 55% of their away matches. (a) Draw a tree diagram to represent the above information. (b) Find the probability that the team wins a match 4. The probability of a pupil from a certain school getting an A in Maths is % of pupils get an A in both Maths and French. (a) Draw a tree diagram (with Maths before French) with as much information as possible on it. (b) Find the probability that he gets an A in Maths but not in French. It is also known that 68% of pupils get an A in French. (c) (d) Find the probability of getting an A in French but not in Maths. Hence complete the tree diagram counters are in a bag of which 5 are green, 4 are blue and 6 are red. Two counters are taken out, one after the other, without replacement. (a) Draw a tree diagram to represent the above information. (b) Find, as fractions in their lowest form, the probability that: (i) At least one is green (ii) The two are different colours. 6. A fair coin is tossed twenty times. Given that the coin showed heads each time what is the probability of the coin showing heads on the next toss?

31 Sheet H c-4 Conditional Probability. The probability that a woman misses her train to work in the morning if there are road works on her journey is 5, otherwise the probability that she misses it is only 5. The probability of there being road works on her journey is 9. (a) Use a tree diagram to represent this information. (b) Find the probability that she catches the train given that there are no road works (c) Find the probability that she misses the train. The probability that a pupil forgets his calculator on Monday is ¼. If he forgets it on Monday then the probability that he forgets it on Tuesday drops to 5. (a) Use a tree diagram to represent as much of this information as possible. (b) Find the probability that he forgets his calculator on both days. The overall probability of him forgetting his calculator on Tuesday is ½. (c) (d) (e) Use part (b) to find the probability that he remembers his calculator on Monday but forgets it on Tuesday. Find the probability that he forgets his calculator on Tuesday given that he remembered it on Monday Complete the tree diagram.. Two competitors A and B shoot at a target. A fires first and the probability that he hits the target is ¼. B fires next and the probability of him hitting the target given that A hit the target is 5. The probability of B hitting the target given that A missed the target is 4 5. (a) Draw a tree diagram to show this information. (b) Find the probability that at least one of the competitors hits the target. (c) Find the probability that only one of them hits the target.

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